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Capital Vol. III Assuming a given wage and working-day, a variable capital, for instance of 100, represents a certain number of employed labourers. It is the index of this number. Suppose £100 are the wages of 100 labourers for, say, one week. If these labourers perform equal amounts of necessary and surplus-labour, if they work daily as many hours for themselves, i.e., for the reproduction of their wage, as they do for the capitalist, i.e., for the production of surplus-value, then the value of their total product = £200, and the surplus-value they produce would amount to £100. The rate of surplus-value, s/v, would = 100%. But, as we have seen, this rate of surplus-value would nonetheless express itself in very different rates of profit, depending on the different volumes of constant capital c and consequently of the total capital C, because the rate of profit = s/C. The rate of surplus-value is 100%: If c = 50, and v = 100, then p' = 100/150 = 66⅔%; c = 100, and v = 100, then p' = 100/200 = 50%; c = 200, and v = 100, then p' = 100/300 = 33⅓%; c = 300, and v = 100, then p' = 100/400 = 25%; c = 400, and v = 100, then p' = 100/500 = 20%. This is how the same rate of surplus-value would express itself under the same degree of labour exploitation in a falling rate of profit, because the material growth of the constant capital implies also a growth — albeit not in the same proportion — in its value, and consequently in that of the total capital. If it is further assumed that this gradual change in the composition of capital is not confined only to individual spheres of production, but that it occurs more or less in all, or at least in the key spheres of production, so that it involves changes in the average organic composition of the total capital of a certain society, then the gradual growth of constant capital in relation to variable capital must necessarily lead to a gradual fall of the general rate of profit, so long as the rate of surplus-value, or the intensity of exploitation of labour by capital, remain the same. Now we have seen that it is a law of capitalist production that its development is attended by a relative decrease of variable in relation to constant capital, and consequently to the total capital set in motion. This is just another way of saying that owing to the distinctive methods of production developing in the capitalist system the same number of labourers, i.e., the same quantity of labour-power set in motion by a variable capital of a given value, operate, work up and productively consume in the same time span an ever-increasing quantity of means of labour, machinery and fixed capital of all sorts, raw and auxiliary materials — and consequently a constant capital of an ever-increasing value. This continual relative decrease of the variable capital vis-a-vis the constant, and consequently the total capital, is identical with the progressively higher organic composition of the social capital in its average. It is likewise just another expression for the progressive development of the social productivity of labour, which is demonstrated precisely by the fact that the same number of labourers, in the same time, i.e., with less labour, convert an ever-increasing quantity of raw and auxiliary materials into products, thanks to the growing application of machinery and fixed capital in general. To this growing quantity of value of the constant capital — although indicating the growth of the real mass of use-values of which the constant capital materially consists only approximately — corresponds a progressive cheapening of products. Every individual product, considered by itself, contains a smaller quantity of labour than it did on a lower level of production, where the capital invested in wages occupies a far greater place compared to the capital invested in means of production. The hypothetical series drawn up at the beginning of this chapter expresses, therefore, the actual tendency of capitalist production. This mode of production produces a progressive relative decrease of the variable capital as compared to the constant capital, and consequently a continuously rising organic composition of the total capital. The immediate result of this is that the rate of surplus-value, at the same, or even a rising, degree of labour exploitation, is represented by a continually falling general rate of profit. (We shall see later [Present edition: Ch. XIV. — Ed.] why this fall does not manifest itself in an absolute form, but rather as a tendency toward a progressive fall.) The progressive tendency of the general rate of profit to fall is, therefore, just an expression peculiar to the capitalist mode of production of the progressive development of the social productivity of labour. This does not mean to say that the rate of profit may not fall temporarily for other reasons. But proceeding from the nature of the capitalist mode of production, it is thereby proved logical necessity that in its development the general average rate of surplus-value must express itself in a falling general rate of profit. Since the mass of the employed living labour is continually on the decline as compared to the mass of materialised labour set in motion by it, i.e., to the productively consumed means of production, it follows that the portion of living labour, unpaid and congealed in surplus-value, must also be continually on the decrease compared to the amount of value represented by the invested total capital. Since the ratio of the mass of surplus-value to the value of the invested total capital forms the rate of profit, this rate must constantly fall. Simple as this law appears from the foregoing statements, all of political economy has so far had little success in discovering it, as we shall see in a later part. [K. Marx, Theorien über den Mehrwert. K. Marx/F. Engels, Werke, Band 26, Teil 2, S. 435-66, 541-43. — Ed.] The economists perceived the phenomenon and cudgelled their brains in tortuous attempts to interpret it. Since this law is of great importance to capitalist production, it may be said to be a mystery whose solution has been the goal of all political economy since Adam Smith, the difference between the various schools since Adam Smith having been in the divergent approaches to a solution. When we consider, on the other hand, that up to the present political economy has been running in circles round the distinction between constant and variable capital, but has never known how to define it accurately; that it has never separated surplus-value from profit, and never even considered profit in its pure form as distinct from its different, independent components, such as industrial profit, commercial profit, interest, and ground-rent; that it has never thoroughly analysed the differences in the organic composition of capital, and, for this reason, has never thought of analysing the formation of the general rate of profit — if we consider all this, the failure to solve this riddle is no longer surprising. We intentionally present this law before going on to the division of profit into different independent categories. The fact that this analysis is made independently of the division of profit into different parts, which fall to the share of different categories of people, shows from the outset that this law is, in its entirety, independent of this division, and just as independent of the mutual relations of the resultant categories of profit. The profit to which we are here referring is but another name for surplus-value itself, which is presented only in its relation to total capital rather than to variable capital, from which it arises. The drop in the rate of profit, therefore, expresses the falling relation of surplus-value to advanced total capital, and is for this reason independent of any division whatsoever of this surplus-value among the various categories. We have seen that at a certain stage of capitalist development, where the organic composition of capital c : v was 50 : 100, a rate of surplus-value of 100% was expressed in a rate of profit of 66⅔%, and that at a higher stage, where c : v was 400 : 100, the same rate of surplus-value was expressed in a rate of profit of only 20%. What is true of different successive stages of development in one country, is also true of different coexisting stages of development in different countries. In an undeveloped country, in which the former composition of capital is the average, the general rate of profit would = 66⅔%, while in a country with the latter composition and a much higher stage of development it would = 20%. The difference between the two national rates of profit might disappear, or even be reversed, if labour were less productive in the less developed country, so that a larger quantity of labour were to be represented in a smaller quantity of the same commodities, and a larger exchange-value were represented in less use-value. The labourer would then spend more of his time in reproducing his own means of subsistence, or their value, and less time in producing surplus-value; consequently, he would perform less surplus-labour, with the result that the rate of surplus-value would be lower. Suppose, the labourer of the less developed country were to work ⅔ of the working-day for himself and ⅓ for the capitalist; in accordance with the above illustration, the same labour-power would then be paid with 133⅓ and would furnish a surplus of only 60⅔. A constant capital of 50 would correspond to a variable capital of 433⅓. The rate of surplus-value would amount to 66⅔ : 133⅓ = 50%, and the rate of profit to 66⅔ : 133⅓, or approximately 36%. Since we have not so far analysed the different component parts of profit, i.e., they do not for the present exist for us, we make the following remarks beforehand merely to avoid misunderstanding: In comparing countries in different stages of development it would be a big mistake to measure the level of the national rate of profit by, say, the level of the national rate of interest, namely when comparing countries with a developed capitalist production with countries in which labour has not yet been formally subjected to capital, although in reality the labourer is exploited by the capitalist (as, for instance, in India, where the ryot manages his farm as an independent producer whose production as such is not, therefore, as yet subordinated to capital, although the usurer may not only rob him of his entire surplus-labour by means of interest, but may also, to use a capitalist term, hack off a part of his wage). This interest comprises all the profit, and more than the profit, instead of merely expressing an aliquot part of the produced surplus-value, or profit, as it does in countries with a developed capitalist production. On the other hand, the rate of interest is, in this case, mostly determined by relations (loans granted by usurers to owners of larger estates who draw ground-rent) which have nothing to do with profit, and rather indicate to what extent usury appropriates ground-rent. As regards countries possessing different stages of development of capitalist production, and consequently capitals of different organic composition, a country where the normal working-day is shorter than another's may have a higher rate of surplus-value (one of the factors which determines the rate of profit). First, if the English ten-hour working-day is, on account of its higher intensity, equal to an Austrian working-day of 14 hours, then, dividing the working-day equally in both instances, 5 hours of English surplus-labour may represent a greater value on the world-market than 7 hours of Austrian surplus-labour. Second, a larger portion of the English working-day than of the Austrian may represent surplus-labour. The law of the falling rate of profit, which expresses the same, or even a higher, rate of surplus-value, states, in other words, that any quantity of the average social capital, say, a capital of 100, comprises an ever larger portion or means of labour, and an ever smaller portion of living labour. Therefore, since the aggregate mass of living labour operating the means of production decreases in relation to the value of these means of production, it follows that the unpaid labour and the portion of value in which it is expressed must decline as compared to the value of the advanced total capital. Or: An ever smaller aliquot part of invested total capital is converted into living labour, and this total capital, therefore, absorbs in proportion to its magnitude less and less surplus-labour, although the unpaid part of the labour applied may at the same time grow in relation to the paid part. The relative decrease of the variable and increase of the constant capital, however much both parts may grow in absolute magnitude, is, as we have said, but another expression for greater productivity of labour. Let a capital of 100 consist of 80c + 20v, and the latter = 20 labourers. Let the rate of surplus-value be 100%, i.e., the labourers work half the day for themselves and the other half for the capitalist. Now let the capital of 100 in a less developed country = 20c + 80v, and let the latter = 80 labourers. But these labourers require 2/3 of the day for themselves, and work only 1/3 for the capitalist. Everything else being equal, the labourers in the first case produce a value of 40, and in the second of 120. The first capital produces 80c + 20v + 20s = 120; rate of profit = 20%. The second capital, 20c + 80v + 40s = 140; rate of profit 40%. In the second case the rate of profit is, therefore, double the first, although the rate of surplus-value in the first = 100%, which is double that of the second, where it is only 50%. But then, a capital of the same magnitude appropriates the surplus-labour of only 20 labourers in the first case, and of 80 labourers in the second case. The law of the progressive falling of the rate of profit, or the relative decline of appropriated surplus-labour compared to the mass of materialised labour set in motion by living labour, does not rule out in any way that the absolute mass of exploited labour set in motion by the social capital, and consequently the absolute mass of the surplus-labour it appropriates, may grow; nor, that the capitals controlled by individual capitalists may dispose of a growing mass of labour and, hence, of surplus-labour, the latter even though the number of labourers they employ does not increase. Take a certain working population of, say, two million. Assume, furthermore, that the length and intensity of the average working-day, and the level of wages, and thereby the proportion between necessary and surplus-labour, are given. In that case the aggregate labour of these two million, and their surplus-labour expressed in surplus-value, always produces the same magnitude of value. But with the growth of the mass of the constant (fixed and circulating) capital set in motion by this labour, this produced quantity of value declines in relation to the value of this capital, which value grows with its mass, even if not in quite the same proportion. This ratio, and consequently the rate of profit, shrinks in spite of the fact that the mass of commanded living labour is the same as before, and the same amount of surplus-labour is sucked out of it by the capital. It changes because the mass of materialised labour set in motion by living labour increases, and not because the mass of living labour has shrunk. It is a relative decrease, not an absolute one, and has, in fact, nothing to do with the absolute magnitude of the labour and surplus-labour set in motion. The drop in the rate of profit is not due to an absolute, but only to a relative decrease of the variable part of the total capital, i.e., to its decrease in relation to the constant part. What applies to any given mass of labour and surplus-labour, also applies to a growing number of labourers, and, thus, under the above assumption, to any growing mass of commanded labour in general, and to its unpaid part, the surplus-labour, in particular. If the working population increases from two million to three, and if the variable capital invested in wages also rises to three million from its former two million, while the constant capital rises from four million to fifteen million, then, under the above assumption of a constant working-day and a constant rate of surplus-value, the mass of surplus-labour, and of surplus-value, rises by one-half, i.e., 50%, from two million to three. Nevertheless, in spite of this growth of the absolute mass of surplus-labour, and hence of surplus-value, by 50%, the ratio of variable to constant capital would fall from 2 : 4 to 3 : 15, and the ratio of surplus-value to total capital would be (in millions) I. 4c + 2v + 2s; C = 6, p' = 33⅓%. II. 15c + 3v + 3s; C = 18, p' = 16⅔%. While the mass of surplus-value has increased by one-half, the rate of profit has fallen by one-half. However, the profit is only the surplus-value calculated in relation to the total social capital, and the mass of profit, its absolute magnitude, is socially equal to the absolute magnitude of the surplus-value. The absolute magnitude of the profit, its total amount, would, therefore, have grown by 50%, in spite of its enormous relative decrease compared to the advanced total capital, or in spite of the enormous decrease in the general rate of profit. The number of labourers employed by capital, hence the absolute mass of the labour set in motion by it, and therefore the absolute mass of surplus-labour absorbed by it, the mass of the surplus-value produced by it, and therefore the absolute mass of the profit produced by it, can, consequently, increase, and increase progressively, in spite of the progressive drop in the rate of profit. And this not only can be so. Aside from temporary fluctuations it must be so, on the basis of capitalist production. Essentially, the capitalist process of production is simultaneously a process of accumulation. We have shown that with the development of capitalist production the mass of values to be simply reproduced, or maintained, increases as the productivity of labour grows, even if the labour-power employed should remain constant. But with the development of social productivity of labour the mass of produced use-values, of which the means of production form a part, grows still more. And the additional labour, through whose appropriation this additional wealth can be reconverted into capital, does not depend on the value, but on the mass of these means of production (including means of subsistence), because in the production process the labourers have nothing to do with the value, but with the use-value, of the means of production. Accumulation itself, however, and the concentration of capital that goes with it, is a material means of increasing productiveness. Now, this growth of the means of production includes the growth of the working population, the creation of a working population, which corresponds to the surplus-capital, or even exceeds its general requirements, thus leading to an over-population of workers. A momentary excess of surplus-capital over the working population it has commandeered, would have a two-fold effect. It could, on the one hand, by raising wages, mitigate the adverse conditions which decimate the offspring of the labourers and would make marriages easier among them, so as gradually to increase the population. On the other hand, by applying methods which yield relative surplus-value (introduction and improvement of machinery) it would produce a far more rapid, artificial, relative over-population, which in its turn, would be a breeding-ground for a really swift propagation of the population, since under capitalist production misery produces population. It therefore follows of itself from the nature of the capitalist process of accumulation, which is but one facet of the capitalist production process, that the increased mass of means of production that is to be converted into capital always finds a correspondingly increased, even excessive, exploitable worker population. As the process of production and accumulation advances therefore, the mass of available and appropriated surplus-labour, and hence the absolute mass of profit appropriated by the social capital, must grow. Along with the volume, however, the same laws of production and accumulation increase also the value of the constant capital in a mounting progression more rapidly than that of the variable part of capital, invested as it is in living labour. Hence, the same laws produce for the social capital a growing absolute mass of profit, and a falling rate of profit. We shall entirely ignore here that with the advance of capitalist production and the attendant development of the productiveness of social labour and multiplication of production branches, hence products, the same amount of value represents a progressively increasing mass of use-values and enjoyments. The development of capitalist production and accumulation lifts labour-processes to an increasingly enlarged scale and thus imparts to them ever greater dimensions, and involves accordingly larger investments of capital for each individual establishment. A mounting concentration of capitals (accompanied, though on a smaller scale, by an increase in the number of capitalists) is, therefore, one of its material requirements as well as one of its results. Hand in hand with it, mutually interacting, there occurs a progressive expropriation of the more or less direct producers. It is, then, natural for the individual capitalists to command increasingly large armies of labourers (no matter how much the variable capital may decrease in relation to the constant), and natural, too, that the mass of surplus-value, and hence profit, appropriated by them, should grow simultaneously with, and in spite of, the fall in the rate of profit. The causes which concentrate masses of labourers under the command of individual capitalists, are the very same that swell the mass of the invested fixed capital, and auxiliary and raw materials, in mounting proportion as compared to the mass of employed living labour. It requires no more than a passing remark at this point to indicate that, given a certain labouring population, the mass of surplus-value, hence the absolute mass of profit, must grow if the rate of surplus-value increases, be it through a lengthening or intensification of the working-day, or through a drop in the value of wages due to an increase in the productiveness of labour, and that it must do so in spite of the relative decrease of variable capital in respect to constant. The same development of the productiveness of social labour, the same laws which express themselves in a relative decrease of variable as compared to total capital, and in the thereby facilitated accumulation, while this accumulation in its turn becomes a starting-point for the further development of the productiveness and for a further relative decrease of variable capital — this same development manifests itself, aside from temporary fluctuations, in a progressive increase of the total employed labour-power and a progressive increase of the absolute mass of surplus-value, and hence of profit. Now, what must be the form of this double-edged law of a decrease in the rate of profit and a simultaneous increase in the absolute mass of profit arising from the same causes? As a law based on the fact that under given conditions the appropriated mass of surplus-labour, hence of surplus-value, increases, and that, so far as the total capital is concerned, or the individual capital as an aliquot part of the total capital, profit and surplus-value are identical magnitudes? Let us take an aliquot part of capital upon which we calculate the rate of profit, e.g., 100. These 100 represent the average composition of the total capital, say, 80c + 20v. We have seen in the second part of this book that the average rate of profit in the various branches of production is determined not by the particular composition of each individual capital, but by the average social composition. As the variable capital decreases relative to the constant, hence the total capital of 100, the rate of profit, or the relative magnitude of surplus-value, i.e., its ratio to the advanced total capital of 100, falls even though the intensity of exploitation were to remain the same, or even to increase. But it is not this relative magnitude alone which falls. The magnitude of the surplus-value or profit absorbed by the total capital of 100 also falls absolutely. At a rate of surplus-value of 100%, a capital of 60c + 40v produces a mass of surplus-value, and hence of profit, amounting to 40; a capital of 70c + 230v a mass of profit of 30; and for a capital of 80c + 20v the profit falls to 20. This falling applies to the mass of surplus-value, and hence of profit, and is due to the fact that the total capital of 100 employs less living labour, and, the intensity of labour exploitation remaining the same, sets in motion less surplus-labour, and therefore produces less surplus-value. Taking any aliquot part of the social capital, i.e., a capital of average composition, as a standard by which to measure surplus-value — and this is done in all profit calculations — a relative fall of surplus-value is generally identical with its absolute fall. In the cases given above, the rate of profit sinks from 40% to 30% and to 20%, because, in fact, the mass of surplus-value, and hence of profit, produced by the same capital falls absolutely from 40 to 30 and to 20. Since the magnitude of the value of the capital, by which the surplus-value is measured, is given as 100, a fall in the proportion of surplus-value to this given magnitude can be only another expression for the decrease of the absolute magnitude of surplus-value and profit. This is, indeed, a tautology. But, as shown, the fact that this decrease occurs at all, arises from the nature of the development of the capitalist process of production. On the other hand, however, the same causes which bring about an absolute decrease of surplus-value, and hence profit, on a given capital, and consequently of the rate of profit calculated in per cent, produce an increase in the absolute mass of surplus-value, and hence of profit, appropriated by the social capital (i.e., by all capitalists taken as a whole). How does this occur, what is the only way in which this can occur, or what are the conditions obtaining in this seeming contradiction? If any aliquot part = 100 of the social capital, and hence any 100 of average social composition, is a given magnitude, for which therefore a fall in the rate of profit coincides with a fall in the absolute magnitude of the profit because the capital which here serves as a standard of measurement is a constant magnitude, then the magnitude of the social capital like that of the capital in the hands of individual capitalists, is variable, and in keeping with our assumptions it must vary inversely with the decrease of its variable portion. In our former illustration, when the percentage of composition was 60c + 40v, the corresponding surplus-value, or profit, was 40, and hence the rate of profit 40%. Suppose, the total capital in this stage of composition was one million. Then the total surplus-value, and hence the total profit, amounted to 400,000. Now, if the composition later = 80c + 20v, while the degree of labour exploitation remained the same, then the surplus-value or profit for each 100 = 20. But since the absolute mass of surplus-value or profit increases, as demonstrated, in spite of the decreasing rate of profit or the decreasing production of surplus-value by every 100 of capital — increases, say, from 400,000 to 440,000, then this occurs solely because the total capital which formed at the time of this new composition has risen to 2,200,000. The mass of the total capital set in motion has risen to 220%, while the rate of profit has fallen by 50%. Had the total capital no more than doubled, it would have to produce as much surplus-value and profit to obtain a rate of profit of 20% as the old capital of 1,000,000 produced at 40%. Had it grown to less than double, it would have produced less surplus-value, or profit, than the old capital of 1,000,000, which, in its former composition, would have had to grow from 1,000,000 to no more than 1,100,000 to raise its surplus-value from 400,000 to 440,000. We again meet here the previously defined law that the relative decrease of the variable capital, hence the development of the social productiveness of labour, involves an increasingly large mass of total capital to set in motion the same quantity of labour-power and squeeze out the same quantity of surplus-labour. Consequently, the possibility of a relative surplus of labouring people develops proportionately to the advances made by capitalist production not because the productiveness of social labour decreases, but because it increases. It does not therefore arise out of an absolute disproportion between labour and the means of subsistence, or the means for the production of these means of subsistence, but out of a disproportion occasioned by capitalist exploitation of labour, a disproportion between the progressive growth of capital and its relatively shrinking need for an increasing population. Should the rate of profit fall by 50%, it would shrink one-half. If the mass of profit is to remain the same, the capital must be doubled. For the mass of profit made at a declining rate of profit to remain the same, the multiplier indicating the growth of the total capital must be equal to the divisor indicating the fall of the rate of profit. If the rate of profit falls from 40 to 20, the total capital must rise inversely at the rate of 20 : 40 to obtain the same result. If the rate of profit falls from 40 to 8, the capital would have to increase at the rate of 8 : 40, or five-fold. A capital of 1,000,000 at 40% produces 400,000, and a capital of 5,000,000 at 8% likewise produces 400,000. This applies if we want the result to remain the same. But if the result is to be higher, then the capital must grow at a greater rate than the rate of profit falls. In other words, for the variable portion of the total capital not to remain the same in absolute terms, but to increase absolutely in spite of its falling in percentage of the total capital, the total capital must grow at a faster rate than the percentage of the variable capital falls. It must grow so considerably that in its new composition it should require more than the old portion of variable capital to purchase labour-power. If the variable portion of a capital = 100 should fall from 40 to 20, the total capital must rise higher than 200 to be able to employ a larger variable capital than 40. Even if the exploited mass of the working population were to remain constant, and only the length and intensity of the working-day were to increase, the mass of the invested capital would have to increase, since it would have to be greater in order to employ the same mass of labour under the old conditions of exploitation after the composition of capital changes. Thus, the same development of the social productiveness of labour expresses itself with the progress of capitalist production on the one hand in a tendency of the rate of profit to fall progressively and, on the other, in a progressive growth of the absolute mass of the appropriated surplus-value, or profit; so that on the whole a relative decrease of variable capital and profit is accompanied by an absolute increase of both. This two-fold effect, as we have seen, can express itself only in a growth of the total capital at a pace more rapid than that at which the rate of profit falls. For an absolutely increased variable capital to be employed in a capital of higher composition, or one in which the constant capital has increased relatively more, the total capital must not only grow proportionately to its higher composition, but still more rapidly. It follows, then, that as the capitalist mode of production develops, an ever larger quantity of capital is required to employ the same, let alone an increased, amount of labour-power. Thus, on a capitalist foundation, the increasing productiveness of labour necessarily and permanently creates a seeming over-population of labouring people. If the variable capital forms just 1/6 of the total capital instead of the former ½, the total capital must be trebled to employ the same amount of labour-power. And if twice as much labour-power is to be employed, the total capital must increase six-fold. Political economy, which has until now been unable to explain the law of the tendency of the rate of profit to fall, pointed self-consolingly to the increasing mass of profit, i.e., to the growth of the absolute magnitude of profit, be it for the individual capitalist or for the social capital, but this was also based on mere platitude and speculation. To say that the mass of profit is determined by two factors — first, the rate of profit, and, secondly, the mass of capital invested at this rate, is mere tautology. It is therefore but a corollary of this tautology to say that there is a possibility for the mass of profit to grow even though the rate of profit may fall at the same time. It does not help us one step farther, since it is just as possible for the capital to increase without the mass of profit growing, and for it to increase even while the mass of profit falls. For 100 at 25% yields 25, and 400 at 5% yields only 20. But if the same causes which make the rate of profit fall, entail the accumulation, i.e., the formation, of additional capital, and if each additional capital employs additional labour and produces additional surplus-value; if, on the other hand, the mere fall in the rate of profit implies that the constant capital, and with it the total old capital, have increased, then this process ceases to be mysterious. We shall see later [K. Marx, Theorien über den Mehrwert. K. Marx/F. Engels, Werke, Band 26, Teil 2,. S. 435-66, 541- 43. — Ed] to what deliberate falsifications some people resort in their calculations to spirit away the possibility of an increase in the mass of profit simultaneous with a decrease in the rate of profit. We have shown how the same causes that bring about a tendency for the general rate of profit to fall necessitate an accelerated accumulation of capital and, consequently, an increase in the absolute magnitude, or total mass, of the surplus-labour (surplus-value, profit) appropriated by it. Just as everything appears reversed in competition, and thus in the consciousness of the agents of competition, so also this law, this inner and necessary connection between two seeming contradictions. It is evident that within the proportions indicated above a capitalist disposing of a large capital will receive a larger mass of profit than a small capitalist making seemingly high profits. Even a cursory examination of competition shows, furthermore, that under certain circumstances, when the greater capitalist wishes to make room for himself on the market, and to crowd out the smaller ones, as happens in times of crises, he makes practical use of this, i.e., he deliberately lowers his rate of profit in order to drive the smaller ones to the wall. Merchants capital, which we shall describe in detail later, also notably exhibits phenomena which appear to attribute a fall in profit to an expansion of business, and thus of capital. The scientific expression for this false conception will be given later. Similar superficial observations result from a comparison of rates of profit in individual lines of business, distinguished either as subject to free competition, or to monopoly. The utterly shallow conception existing in the minds of the agents of competition is found in Roscher, namely, that a reduction in the rate of profit is "more prudent and humane". [Roscher, Die Grundlage der Nationalökonomie, 3 Auflage, 1858, 108, S. 192. — Ed.] The fall in the rate of profit appears in this case as an effect of an increase in capital and of the concomitant calculation of the capitalist that the mass of profits pocketed by him will be greater at a smaller rate of profit. This entire conception (with the exception of Adam Smith's, which we shall mention later) [K. Marx, Theorien über den Mehrwert. K. Marx/F. Engels, Werke, Band 26, Teil 2, S. 214-28. — Ed.] rests on an utter misapprehension of what the general rate of profit is, and on the crude notion that prices are actually determined by adding a more or less arbitrary quota of profit to the true value of commodities. Crude as these ideas are, they arise necessarily out of the inverted aspect which the immanent laws of capitalist production represent in competition. The law that a fall in the rate of profit due to the development of productiveness is accompanied by an increase in the mass of profit, also expresses itself in the fact that a fall in the price of commodities produced by a capital is accompanied by a relative increase of the masses of profit contained in them and realised by their sale. Since the development of the productiveness and the correspondingly higher composition of capital sets in motion an ever-increasing quantity of means of production through a constantly decreasing quantity of labour, every aliquot part of the total product, i.e., every single commodity, or each particular lot of commodities in the total mass of products, absorbs less living labour, and also contains less materialised labour, both in the depreciation of the fixed capital applied and in the raw and auxiliary materials consumed. Hence every single commodity contains a smaller sum of labour materialised in means of production and of labour newly added during production. This causes the price of the individual commodity to fall. But the mass of profits contained in the individual commodities may nevertheless increase if the rate of the absolute or relative surplus-value grows. The commodity contains less newly added labour, but its unpaid portion grows in relation to its paid portion. However, this is the case only within certain limits. With the absolute amount of living labour newly incorporated in individual commodities decreasing enormously as production develops, the absolute mass of unpaid labour contained in them will likewise decrease, however much it may have grown as compared to the paid portion. The mass of profit on each individual commodity will shrink considerably with the development of the productiveness of labour, in spite of a growth in the rate of surplus-value. And this reduction, just as the fall in the rate of profit, is only delayed by the cheapening of the elements of constant capital and by the other circumstances set forth in the first part of this book, which increase the rate of profit at a given, or even falling, rate of surplus-value. That the price of individual commodities whose sum makes up the total product of capital falls, means simply that a certain quantity of labour is realised in a larger quantity of commodities, so that each individual commodity contains less labour than before. This is the case even if the price of one part of constant capital, such as raw material, etc., should rise. Outside of a few cases (for instance, if the productiveness of labour uniformly cheapens all elements of the constant, and the variable, capital), the rate of profit will fall, in spite of the higher rate of surplus-value, 1) because even a larger unpaid portion of the smaller total amount of newly added labour is smaller than a smaller aliquot unpaid portion of the former larger amount and 2) because the higher composition of capital is expressed in the individual commodity by the fact that the portion of its value in which newly added labour is materialised decreases in relation to the portion of its value which represents raw and auxiliary material, and the wear and tear of fixed capital. This change in the proportion of the various component parts in the price of individual commodities, i.e., the decrease of that portion of the price in which newly added living labour is materialised, and the increase of that portion of it in which formerly materialised labour is represented, is the form which expresses the decrease of the variable in relation to the constant capital through the price of the individual commodities. Just as this decrease is absolute for a certain amount of capital, say of 100, it is also absolute for every individual commodity as an aliquot part of the reproduced capital. However, the rate of profit, if calculated merely on the elements of the price of an individual commodity, would be different from what it actually is. And for the following reason: [The rate of profit is calculated on the total capital invested, but for a definite time, actually a year. The rate of profit is the ratio of the surplus-value, or profit, produced and realised in a year, to the total capital calculated in per cent. It is, therefore, not necessarily equal to a rate of profit calculated for the period of turnover of the invested capital rather than for a year. It is only if the capital is turned over exactly in one year that the two coincide. On the other hand, the profit made in the course of a year is merely the sum of profits on commodities produced and sold during that same year. Now, if we calculate the profit on the cost-price of commodities, we obtain a rate of profit = p/k in which p stands for the profit realised during one year, and k for the sum of the cost-prices of commodities produced and sold within the same period. It is evident that this rate of profit p/k will not coincide with the actual rate of profit p/C, mass of profit divided by total capital, unless k = C, that is, unless the capital is turned over in exactly one year. Let us take three different conditions of an industrial capital. I. A capital of £8,000 produces and sells annually 5,000 pieces of a commodity at 30s. per piece, thus making an annual turnover of £7,500. It makes a profit of 10s. on each piece, or £2,500 per year. Every piece, then, contains 20s. advanced capital and 10s. profit, so that the rate of profit per piece is 10/20 = 50%. The turned-over sum of £7,500 contains £5,000 advanced capital and £2,500 profit. Rate of profit per turnover, p/k, likewise 50%. But calculated on the total capital the rate of profit p/C = 2,500/8,000 = 31¼% II. The capital rises to £10,000. Owing to increased productivity of labour it is able to produce annually 10,000 pieces of the commodity at a cost-price of 20s. per piece. Suppose the commodity is sold at a profit of 4s., hence at 24s. per piece. In that case the price of the annual product = £12,000, of which £10,000 is advanced capital and £2,000 is profit. The rate of profit p/k = 4/20 per piece, and 2,000/10,000 for the annual turnover, or in both cases = 20%. And since the total capital is equal to the sum of the cost-prices, namely £10,000, it follows that p/C, the actual rate of profit, is in this case also 20%. III. Let the capital rise to £15,000 owing to a constant growth of the productiveness of labour, and let it annually produce 30,000 pieces of the commodity at a cost-price of 13s. per piece, each piece being sold at a profit of 2s., or at 15s. The annual turnover therefore = 30,000×15s. = £22,500, of which £19,500 is advanced capital and £3,000 profit. The rate of profit p/k then = 2/13 = 3,000/19,500 = 15 5/13%. But p/C = 3,000/15,000 = 20%. We see, therefore, that only in case II, where the turned-over capital-value is equal to the total capital, the rate of profit per piece, or per total amount of turnover, is the same as the rate of profit calculated on the total capital. In case I, in which the amount of the turnover is smaller than the total capital, the rate of profit calculated on the cost-price of the commodity is higher; and in case III, in which the total capital is smaller than the amount of the turnover, it is lower than the actual rate calculated on the total capital. This is a general rule. In commercial practice, the turnover is generally calculated inaccurately. It is assumed that the capital has been turned over once as soon as the sum of the realised commodity-prices equals the sum of the invested total capital. But the capital can complete one whole turnover only when the sum of the cost-prices of the realised commodities equals the sum of the total capital. — F.E.] This again shows how important it is in capitalist production to regard individual commodities, or the commodity-product of a certain period, as products of advanced capital and in relation to the total capital which produces them, rather than in isolation, by themselves, as mere commodities. The rate of profit must be calculated by measuring the mass of produced and realised surplus-value not only in relation to the consumed portion of capital reappearing in the commodities, but also to this part plus that portion of unconsumed but applied capital which continues to operate in production. However, the mass of profit cannot be equal to anything but the mass of profit or surplus-value, contained in the commodities themselves, and to be realised by their sale. If the productivity of industry increases, the price of individual commodities falls. There is less labour in them, less paid and unpaid labour. Suppose, the same labour produces, say, triple its former product. Then ⅔ less labour yields individual product. And since profit can make up but a portion of the amount of labour contained in an individual commodity, the mass of profit in the individual commodity must decrease, and this takes place within certain limits, even if the rate of surplus-value should rise. In any case, the mass of profit on the total product does not fall below the original mass of profit so long as the capital employs the same number of labourers at the same degree of exploitation. (This may also occur if fewer labourers are employed at a higher rate of exploitation.) For the mass of profit on the individual product decreases proportionately to the increase in the number of products. The mass of profit remains the same, but it is distributed differently over the total amount of commodities. Nor does this alter the distribution between the labourers and capitalists of the amount of value created by newly added labour. The mass of profit cannot increase so long as the same amount of labour is employed, unless the unpaid surplus-labour increases, or, should intensity of exploitation remain the same, unless the number of labourers grows. Or, both these causes may combine to produce this result. In all these cases — which, however, in accordance with our assumption, presuppose an increase of constant capital as compared to variable, and an increase in the magnitude of total capital — the individual commodity contains a smaller mass of profit and the rate of profit falls even if calculated on the individual commodity. A given quantity of newly added labour materialises in a larger quantity of commodities. The price of the individual commodity falls. Considered abstractly the rate of profit may remain the same, even though the price of the individual commodity may fall as a result of greater productiveness of labour and a simultaneous increase in the number of this cheaper commodity if, for instance, the increase in productiveness of labour acts uniformly and simultaneously on all the elements of the commodity, so that its total price falls in the same proportion in which the productivity of labour increases, while, on the other hand, the mutual relation of the different elements of the price of the commodity remains the same. The rate of profit could even rise if a rise in the rate of surplus-value were accompanied by a substantial reduction in the value of the elements of constant, and particularly of fixed, capital. But in reality, as we have seen, the rate of profit will fall in the long run. In no case does a fall in the price of any individual commodity by itself give a clue to the rate of profit. Everything depends on the magnitude of the total capital invested in its production. For instance, if the price of one yard of fabric falls from 3s. to 1⅔s., if we know that before this price reduction it contained 1⅔s. constant capital, yarn, etc., ⅔s. wages, and ⅔s. profit, while after the reduction it contains 1s. constant capital, $#8531s. wages, and ⅓s. profit, we cannot tell if the rate of profit has remained the same or not. This depends on whether, and by how much, the advanced total capital has increased, and how many yards more it produces in a given time. The phenomenon, springing from the nature of the capitalist mode of production, that increasing productivity of labour implies a drop in the price of the individual commodity, or of a certain mass of commodities, an increase in the number of commodities, a reduction in the mass of profit on the individual commodity and in the rate of profit on the aggregate of commodities, and an increase in the mass of profit on the total quantity of commodities — this phenomenon appears on the surface only in a reduction of the mass of profit on the individual commodity, a fall in its price, an increase in the mass of profit on the augmented total number of commodities produced by the total social capital or an individual capitalist. It then appears as if the capitalist adds less profit to the price of the individual commodity of his own free will, and makes up for it through the greater number of commodities he produces. This conception rests upon the notion of profit upon alienation, which, in its turn, is deduced from the conception of merchant capital. We have previously seen in Book I (4 and 7 Abschnitt) [English edition: Parts IV and VII. — Ed.] that the mass of commodities growing along with the productivity of labour and the cheapening of the individual commodity as such (as long as these commodities do not enter the price of labour-power as determinants) — that this does not affect the proportion between paid and unpaid labour in the individual commodity, in spite of the falling price. Since all things appear distorted, namely, reversed in competition, the individual capitalist may imagine: 1) that he is reducing his profit on the individual commodity by cutting its price, but still making a greater profit by selling a larger quantity of commodities; 2) that he fixes the price of the individual commodities and that he determines the price of the total product by multiplication, while the original process is really one of division (see Book I, Kap. X, S. 281 [English edition: Ch. XII. — Ed]), and multiplication is only correct secondarily, since it is based on that division. The vulgar economist does practically no more than translate the singular concepts of the capitalists, who are in the thrall of competition, into a seemingly more theoretical and generalised language, and attempt to substantiate the justice of those conceptions. The fall in commodity-prices and the rise in the mass of profit on the augmented mass of these cheapened commodities is, in fact, but another expression for the law of the falling rate of profit attended by a simultaneously increasing mass of profit. The analysis of how far a falling rate of profit may coincide with rising prices no more belongs here than that of the point previously discussed in Book I (S. 280-81 [English edition: Ch. XII. — Ed.]), concerning relative surplus-value. A capitalist working with improved but not as yet generally adopted methods of production sells below the market-price, but above his individual price of production; his rate of profit rises until competition levels it out. During this equalisation period the second requisite, expansion of the invested capital, makes its appearance. According to the degree of this expansion the capitalist will be able to employ a part of his former labourers, actually perhaps all of them, or even more, under the new conditions, and hence to produce the same, or a greater, mass of profit. 1. "We should also expect that, however the rate of the profits of stock might diminish in consequence of the accumulation of capital on the land and the rise of wages, yet the aggregate amount of profits would increase. Thus, supposing that, with repeated accumulations of £100,000, the rate of profit should fall from 20 to 19, to 18, to 17%, a constantly diminishing rate, we should expect that the whole amount of profits received by those successive owners of capital would be always progressive; that it would be greater when the capital was £200,000, than when £100,000; still greater when £300,000; and so on, increasing, though at a diminishing rate, with every increase of capital. This progression, however, is only true for a certain time; thus 19% on £200,000 is more than 20% on £100,000; again 18% on £300,000 is more than 19% on £200,000; but after capital has accumulated to a large amount, and profits have fallen, the further accumulation diminishes the aggregate of profits. Thus, suppose the accumulation should be £1,000,000, and the profits 7%, the whole amount of profits will be £70,000; now if an addition of £100,000 capital be made to the million, and profits should fall to 6%, £66,000 or a diminution of £4,000 will be received by the owners of the stock, although the whole amount of stock will be increased from £1,000,000 to £1,100,000." — Ricardo, Political Economy, Chap. VI (Works, ed. by MacCulloch, 1852, pp. 68-69). — The fact is, that the assumption has here been made that the capital increases from 1,000,000 to 1,100,000, that is, by 10%, while the rate of profit falls from 7 to 6, hence by 14 2/7 %. Hinc illae lacrimae! ['Thus these tears' - Publius, Terence, Andria, Act I, Scene 1. — Ed.]

Rl Circuit Charging And Discharging Since this lasts for only a "short" time, this is known as a transient effect. 1 RL circuit 222 7. Note that since the turn off current flows through both RH and RL, adjustment to the RL value is required to ensure the desired turn-off transition time (see 3. Time Constant (τ): A measure of time required for certain changes in voltages and currents in RC and RL circuits. The line current is a sinusoidal waveform while the reactor current id, is a rectified current. The circuits for charging and discharging the capacitor were set up as specified in the lab manual. There will be a transient interval while the voltages and currents in the. sa mga students ni Sir Jemay :D. Since these charging and discharging processes occur very rapidly, a convenient way to study these processes is using the oscilloscope since time scales are now milliseconds rather than seconds. Comparison of RC and RL Circuits: The comparison between RC and RI circuits are given as under: RC circuits occupy small space as compared to RL circuit. QUESTION: I have a 2004 5th wheel. 3K and C = 100nf. C R! Charging Discharging Discharging For the discharging case, applying the loop rule to the circuit gives: Chapter H - Inductance and Transient Circuits 3. The time required to charge a capacitor to 63 percent (actually 63. Charging and Discharging RC Circuit Under Riemann-Liouville and Caputo Fractional Derivatives Amr M. AbdelAty 1, Ahmed G. The time constant τ is given by τ = L / R. , an inductor behaves like a short circuit in DC conditions as one would expect from a highly conducting coil. Answer to: An uncharged capacitor and a resistor are connected in series to a source of emf. As time elapses toward time t 1, there is a continuous decrease in current flowing into the capacitor. First consider what happens with the resistor and the. (1) (Charging Circuit equation) Where R is the resistance value, and C is the capacitance. To study charging and discharging current characteristics in an RC circuit. While discharging, the opposite happens. Observe the response of the circuit for the following three cases and record the results. Derivation for voltage across a charging and discharging capacitor by admin · Published March 21, 2017 · Updated February 3, 2019 Here derives the expression to obtain the instantaneous voltage across a charging capacitor as a function of time, that is V (t). In this section we see how to solve the differential equation arising from a circuit consisting of a resistor and a capacitor. For DC a capacitor acts like an. The time dependence of the potential di erence V(t) for the charging and discharging process is shown in Figure 2. 2 Introduction and Test Circuits Inductors and capacitors have the ability to store energy. t1 is the time when the capacitor reaches full charge or discharge. edu Abstract This paper describes a small, compact circuit that captures the temporal and adaptation properties both of the photoreceptor and. Generally, ripple is undesirable, thus the smaller the ripple, the better the filtering action. Once the voltage supply becomes superior to the voltage of the capacitor, the capacitor gets charging. Search: Search To study lissajous figures experiment pdf. If the RL circuit is shorted by another switch, Figure 7 view A, the stored energy in the field of L instantly develops a voltage, view B. Installing Fixed - Mounted Circuit Breakers 11 Installing the Electronic Trip Unit 12-14 Inserting/Removing Rating Pug 14-15 Operating Instructions Part One - Operating the SB Encased Systems Breaker Manually Charging the Stored Energy Mechanism 17 Discharging the Stored Energy Mechanism 18 Locally Closing the Circuit Breaker 18. The red represents the charging while the blue represents the discharging. Transient RC series circuits charging a capacitor, charging and discharging a capacitor in a RC circuit: Transient Response of RC Circuit Transient Response of RC Circuit. To change the voltage suddenly, a function generator will be used. If a switch is added to the circuit but is open, no current flows. V t Figure 4: Square wave applied to RC circuit in blue and voltage across capacitor in dashed red vs. Average values and RMS. It is important to determine the. RC Circuit Analysis No. Figure below shows a circuit containing resistance R and inductance L connected in series combination through a battery of constant emf E through a two way switch S; To distinguish the effects of R and L,we consider the inductor in the circuit as resistance less and resistance R as non-inductive. RL Circuits - Discharging e R I I e or R K I R At t 0, I I I K' Ke (K e ) t K' L R n I - dt L R-I dI IR LdI -IRdt dt dI 0 L t L R t - L R - 0 0 0 t L R - t L R - L 0 t L R - R 0 CR t - 0-I Re d t d I V L V IR I Re I I e Integration constant V R + V L = 0 Current L R Note special switch charge discharge. Introduction to Electric Circuits Lab Manual. Analysis of a Simple R-L Circuit and Inductor Behavior Analysis of a Simple R-L Circuit with DC Supply: The circuit shown in Figures-1 is a simple R-L circuit (it has one simple resistor & inductor connected in series with a voltage supply of 2V); Though it is a simple circuit but if you will analyze it, your Electrical Engineering basics will be enhanced. Covers through text Sec. Cite this Video. The potential rises exponentially from 0 V to the final value of U0. RC Circuits Charging CapacitorThis means, at t = 0 seconds, charge incapacitor is zero. The LED drive current is calculated by the following equation. 5, Charging a Capacitor, Voltage and Current, Worked Example RC Circuits No. In a resistor-capacitor charging circuit, capacitor voltage goes from nothing to full source voltage while current goes from maximum to zero, both variables changing most rapidly at first, approaching their final values slower and slower as time goes on. In accordance with the present invention as described above, by selecting attack charging resistors RL' and RL" and release discharging resistor R100, it is possible to independently set the attack time constants in response to positive or negative input pulses and also the release time constant upon cessation of the input pulses of a single. Operation of Sync-tip-clamp Input circuit will be explained. GSIs will review material in discussion sections prior to the exam. The capacitor would then be charging, discharging, charging, discharging, etc. What you would need to do I believe (from my head) is build a simple RL circuit. (John 2010) developed a model for charging and discharging of a capacitor. When an inductor is charging, the energy is stored in the magnetic field and that stored energy is used to drive the current while discharging of the inductor. Read more about units of capacitance and discharging a capacitor. To explore the time dependent behavior of RC and RL Circuits. RC Circuit Definition: The combination of a pure resistance R in ohms and pure capacitance C in Farads is called RC circuit. In RL series circuit, during the inductor charging phase, the voltage across the inductor gradually decrease to zero and the current through the inductor goes to the maximum in five-times constant (5 taus). If the RL circuit is shorted by another switch, Figure 7 view A, the stored energy in the field of L instantly develops a voltage, view B. Other features of this circuit are a variable regulated DC voltage output (0-12V), voltage display panel meter, provision to measure charging/discharging current, ammeter, and micro soldering iron. I know an inductor resists. 1 RC Circuit Analysis No. The voltage VA2 is now given by: VA2= Vmin. Simscape™ Electrical™ Specialized Power Systems allows you to build and simulate electrical circuits containing linear and nonlinear elements. Assume that the switch, S is open until it is closed at a time t = 0, and then remains permanently closed producing a "step response" type voltage input. (See Figure 4. Salient Features * Topics Have Been Carefully Selected To Give A Flavour Of Computational Techniques In The Context Of A Wide Range Of Physics Problems. Inductor transient response. The Series RC Circuit and the Oscilloscope We shall use the oscilloscope to study charging and discharging of a capacitor in a simple RC circuit similar to the one shown in Fig. Since these charging and discharging processes occur very rapidly, a convenient way to study these processes is using the oscilloscope since time scales are now milliseconds rather than seconds. 0 available with a max astonishing power of 36W. Simple Capacitor Discharge High Voltage Generator Circuit Diagram Stepdown transformer T1 drops the incoming line voltage to approximately 48 Vac which is rectified by diode D1; the resultant de charges capacitor C1-through current limiting resistor Rl-to a voltage level preset by R4. Inserting. Discharging Circuits When the switch is closed in the circuit shown below, charges immediately start flowing off of the plates of the capacitor. In Figure 3, the Capacitor is connected to the DC Power Supply and Current flows through the circuit. 3 Quantity analysis of the RL energy-storing process 224. After completing this chapter, you will be able to: 1) understand the first-order circuits and concepts of the step response and source-free response of the circuits 2) understand the initial conditions in the switching circuit 3) understand the concepts of the transient and steady states of RL and RC circuits 4) determine the charging/discharging process in an RC circuit 5) determine the. There are some similarities between the RL circuit and the RC circuit, and some important differences. RL circuits In an RC circuit, while charging, Q = CV and the loop rule mean: • charge increases from 0 to CE • current decreases from E/R to 0 • voltage across capacitor increases from 0 to E In an RL circuit, while “charging” (rising current), emf = Ldi/dt and the loop rule mean: • magnetic field increases from 0 to B. For a resistor-capacitor circuit, the time constant (in seconds) is calculated from the product (multiplication) of resistance in ohms and. Charging a supercapacitor through a standard resistance can take a long time due to the RC time constant. This is the subject of Chapter 9. The RL DifferentiatorAn RL differentiator is also a circuit that approximates the mathematical process of differentiation. Answer to: An uncharged capacitor and a resistor are connected in series to a source of emf. If the mechanical energy is then converted to electricity, the machine is called a wind generator, wind turbine, wind power unit (WPU), wind energy converter (WEC), or aero generator. The response of charging and discharging of RL circuit is not better than. RL Vref The length of time that the green LED is illuminated:. 4 Current Flowing through a Series RLC Circuit 5. This pulse train has a period T and frequency f=1/T, and amplitude V0. Natural Commutation. Note that when the capacitor is discharging the charge is decreasing and thus the current is negative. If we use a resistance in series, instead of the inductor as the filter, these drawbacks can be overcome. We write equations for across the inductor. When voltage is. 3 Quantity analysis of the RL energy-storing process 224. 8 Efficiency 13. RLC Circuit Confusions Textbook Method for LCR Series Circuit while Discharging – The Wrong One Voltage across an inductor(L) = L(dI/dt), where I(t) = Current through the Inductor at time t. Analysis of a Simple R-L Circuit and Inductor Behavior Analysis of a Simple R-L Circuit with DC Supply: The circuit shown in Figures-1 is a simple R-L circuit (it has one simple resistor & inductor connected in series with a voltage supply of 2V); Though it is a simple circuit but if you will analyze it, your Electrical Engineering basics will be enhanced. Search: Search To study lissajous figures experiment pdf. The formulas on this page are associated with a series RLC circuit discharge since this is the primary model for most high voltage and pulsed power discharge circuits. Be sure the lab instructor or TA checks the circuit before you turn the power on! Note that the probes to Channel A of the Science Workshop Interface are set to measure voltage (which is proportional to charge) across the capacitor. The bigger τ is the longer it takes for the circuit energy to discharge. « Last Edit:. First consider what happens with the resistor and the. An Inductor opposes CHANGES in current flow. LR circuits work in a similar manner. Charging a capacitor means the accumulation of charge over the plates of the capacitor, where discharging is a completely opposite process. The RC Circuit The RC circuit is the electrical circuit consisting of a resistor of resistance R, a capacitor of capacitance C and a voltage source arranged in series. JoVE Science Education Database. RL Circuits - Discharging e R I I e or R K I R At t 0, I I I K' Ke (K e ) t K' L R n I - dt L R-I dI IR LdI -IRdt dt dI 0 L t L R t - L R - 0 0 0 t L R - t L R - L 0 t L R - R 0 CR t - 0-I Re d t d I V L V IR I Re I I e Integration constant V R + V L = 0 Current L R Note special switch charge discharge. An RC circuit is composed of a resistor and a capacitor, while an RL circuit is composed of a resistor and an inductor. 29 A when. Charging (and discharging) of capacitors follows an exponential law. • Such circuits (usually referred to as RL, RC, or RLC circuits) are of great interest in electrical. conducting till the peak are reached and this cycle continues. Look into the subject as per the topics and aware of marks, pattern,. The capacitor in the circuit is not charged fully, so the charging of this does not occur instantly. Discharging Time Constants of an RC Circuit. Consider a simple RL circuit in which resistor, R and inductor, L are connected in series with a voltage supply of V volts. Capacitor Charging and Discharging Part 1 A capacitor charging and discharging phenomenon is one of the most important factor in Analog electronics. The time dependence of the potential di erence V(t) for the charging and discharging process is shown in Figure 2. An RC series circuit contains a voltage source with a resistor and a capacitor in series. Purpose: Learn the capacitor are components that storing the energy via the electric field, and we will consider charging and discharging a capacitor. PHYS 142 FORMULA SHEET Nicholas Salloum Electromotive Force and Power EMF of ideal source: 𝜀= å â æ æ EMF of source with internal resistance: =𝜀−. Decoupling Capacitors and RF Networks. To understand the time constant in an RC circuit and how it can be changed. , 50 ms) This is the basis of clamper circuit operation. The experimental case in which the capacitor is charging in the circuit will be referred to as the charging capacitor case. Illustration of the Experiment. A basic RL differentiator circuit is an inductor in series with a resistor and the source. Let's put an inductor (i. Compute RC from component values. We write equations for across the inductor. Time Constant (t): A measure of time required for certain changes in voltages and currents in RC and RL circuits. 0 µF, and = 28 V. This size of this resistance is such that the associated power loss is equal to the sum of the insulation and. Some responses. Kirkwood Community College Course Syllabus. When voltage is applied to the capacitor, the charge builds up in the capacitor and the current drops off to zero. charging times τ and 2τ and discharging times (3τ and 4τ). EE 201 RL transient - 1 RL transients Circuits having inductors: • At DC - inductor is a short circuit, just another piece of wire. Theoretically, the time constant is given by the product of the resistance and capacitance in the circuit, RC. Each USB port equals an original charger of the device and offers a full speed charging experience for both devices simultaneously ranging from Android/iOS smartphones and tablets to other USB devices, while the slimline design avoids blocking other outlets. 555 timer is a medium-sized integrated circuit that combines analog functions and logic functions. Charging the capacitor and timing how long it took to reach our target value of volts yielded the experimental value of RC. ppt), PDF File (. These circuits provide the sufficient current for charging of battery. Rc and rl differentiator and integrator circuit 1. is -V, the capacitor is discharging. Where: rr rL L L Li it SL S L t,,(). CHARGE AND DISCHARGE OF A CAPACITOR Figure 5. Voltage (solid line) leads current through an inductor (a) and voltage lags current through a capacitor (b). A Capacitor is a passive device that stores energy in its Electric Field and returns energy to the circuit whenever required. RC Time Constant Calculator If a voltage is applied to a capacitor of Value C through a resistance of value R, the voltage across the capacitor rises slowly. Ninth Edition. Review of charging and discharging in RC Circuits (an enlightened approach) • Before we continue with formal circuit analysis - lets review RC circuits • Rationale: Every node in a circuit has capacitance to ground, like it or not, and it’s the charging of these capacitances that limits real circuit performance (speed). Charging of a Capacitor with Different Time Constants Charging and Discharging of a Capacitor 5. 5, Charging a Capacitor, Voltage and Current, Worked Example RC Circuits No. τ is the time needed for the Transient Response to decay by a factor of 1/e. Also, the antenna size D omax. RC Circuit Analysis No. A switch alternates rapidly, switching the inductor between charging and discharging states. The circuits for charging and discharging the capacitor were set up as specified in the lab manual. 2 - Single Loop RL and RC Discharging (Release) Circuits; Section 3. 3, topics of HW 4. Circuit designers must be careful to ensure that the period of the square wave gives sufficient time for the capacitor to fully charge and discharge. Ahmed , Mariam Faied4 1Engineering Mathematics and. Charging capacitor Discharging capacitor The RL circuits. Parameters of two port network 5. The purpose of this lab is to understand the concept of RC charging and discharging circuit and then see how a real capacitor works in the circuit. The results are also similar. Charging and Discharging the Capacitor. RC differentiator RC differentiator, illustrates a simple RC differentiator, the charge stored by the capacitor: RC integrator RC integrator: RC time constant this tutorial illustrates how the amount time required to charge and discharge a capacitor is a very important factor in the design of electrical circuits: RL-circuit RL-circuit. With Vf being 0 and Vi=V₀ when discharging, and the opposite when charging. RV BATTERIES. 5, Charging a Capacitor, Voltage and Current, Worked Example RC Circuits No. The currents used for quick charge and trickle charge are detemined by the voltage of the terminals Iset1 and Iset2 and current sense resistor R4 (Application in page 9). A first-order RL circuit is composed of one resistor and one inductor and is the simplest type of RL circuit. Collect your data in the table below. In an RL circuit, the impedance is determined by both the resistance and the inductive reactance combined The impedance of an RL circuit varies directly with frequency The phase angle (θ) if a series RL circuit varies directly with frequency In an RL lag network, the output voltage lags the input voltage in phase. It is not possible to fabricate inductors on the surface of semiconductor chip. 29 A when. Physics II. Part 1: Charging & Discharging With Data Studio you can knock out both the classic cases of charging and discharging in "one fell swoop". allow you to optimise your welder design and parameters for a given weld, 3. R1 / (R1 + R2). RL Circuits RC Circuits LC (RLC) Circuits and Electromagnetic Oscillations AC Circuits with AC Source Phasors Phaser Turn on RL Circuit (reminder) Note: the time constant is τ=L/R I= V 0 R (1−e − t L/R) Turn off RL Circuit (reminder) Note: the time constant is τ=L/R I=I 0 e − t L/R Charging a Capacitor in RC Circuit Charging q(t)=εC(1. First consider what happens with the resistor and the. As time elapses toward time t 1, there is a continuous decrease in current flowing into the capacitor. Charging capacitor Discharging capacitor The RL circuits. Charging a capacitor •Although no charge actually passes between the capacitor plates, it acts just like a current is flowing through it. To understand transient behavior of capacitor let us draw a RC circuit as shown below,. The transient behavior of a circuit with a battery, a resistor and a capacitor is governed by Ohm's law, the voltage law and the definition of capacitance. It may be due to the influence of different electrodes while charging and discharging. This physics video tutorial explains how to solve RC circuit problems with capacitors and resistors. 1 as a function of time if the. I was thinking about the discharging mechanism of an inductor and I have a question about it. It indicates the response time (how fast you can up a current) of the RC circuit. To study charging and discharging current characteristics in an RC circuit. LC Circuits Consider an electrical circuit consisting of an inductor, of inductance , connected in series with a capacitor, of capacitance. RC Circuits Charging Capacitor2nd equation means. Lab 6: RC Transient Circuits If the time constant is very large relative to the half-period (T/2) of the input pulse, the circuit does not even come close to charging before the pulse falls again. In this section we see how to solve the differential equation arising from a circuit consisting of a resistor and a capacitor. After one time constant 𝜏𝜏, the current is 1/𝑒𝑒 of its initial value. Using SPICE to simulate circuits PRELAB: Find the voltage across the capacitor in Fig. Analyze the discharging of a capacitor through an inductor. To explore the time dependent behavior of RC and RL Circuits. So now we come to the Boostcaps, to trick the dummy charger into thinking it is charging a Lead Acid battery. Inductors have the exact opposite characteristics of capacitors. The RC Circuit The RC circuit is the electrical circuit consisting of a resistor of resistance R, a capacitor of capacitance C and a voltage source arranged in series. , a coil with an inductance L) in series with a battery of emf ε and a resistor of resistance R. Analyze an RL circuit hooked up to a battery. In the first phase (Fig. Build and Simulate a Simple Circuit Introduction. RC, RL and LC circuits are essential building blocks in many circuit applications. circuit will settle into a new DC state, where the capacitors are again open-circuits. When the rms output voltage of a bridge full-wave rectifier is 20V, the peak inverse voltage across the diodes is (neglecting the diode drop). The filter made up of capacitor and resistor is known as capacitor filter. Charging and discharging. Then for a RC discharging circuit that is initially fully charged, the voltage across the capacitor after one time constant, 1T, has dropped by 63% of its initial value which is 1 – 0. An RL Circuit without a Source of emf. CONTENT (i) Capacitance in series and parallel. Circuit Diagram Battery Charger Using Scr. The plots should start from 0s and end at 1s. RL circuits In an RC circuit, while charging, Q = CV and the loop rule mean: • charge increases from 0 to CE • current decreases from E/R to 0 • voltage across capacitor increases from 0 to E In an RL circuit, while "charging" (rising current), emf = Ldi/dt and the loop rule mean: • magnetic field increases from 0 to B. The circuit is a 30 Volt circuit, located on the left vertical side, followed on by a 6 ohm resistor, then the switch on […]. 0 V, C = 24. There will be a transient interval while the voltages and currents in the. Through this process, the time duration during which Ft is to be noted here that the capacitor C gets charged to the peak because there is no resistance (except the negligible forward resistance of diode) in the charging path. The time constant for a circuit having a 100 microfarad capacitor in series with a 470K resistor is:. Conclusion & Discussion At the center of this experiment was the capacitor and its properties. As Vin decreases, Q2 is on and pulls a current out of RL. Built-in reverse polarity protection: While the charger is connected, each charging channel has reverse polarity protection, the charging light is still green and prevent from any short circuit. The RC circuit equation defines the RC time constant as the product of resistance and capacitance. Compute RC from component values. Capacitance, Charging and Discharging of a Capacitor Capacitance is the ability of a capacitor to store maximum electrical charge in its body. Analysis of a Simple R-L Circuit and Inductor Behavior Analysis of a Simple R-L Circuit with DC Supply: The circuit shown in Figures-1 is a simple R-L circuit (it has one simple resistor & inductor connected in series with a voltage supply of 2V); Though it is a simple circuit but if you will analyze it, your Electrical Engineering basics will be enhanced. The reasons for this analysis are to: 1. Module 2 Electricity Sources Resources available Module 3 Resistors Resources available. must happen to the current in the circuit? Explain. As you may know, it takes infinite time to charge a capacitor. In lecture, we used calculus to show that V c changes with time according to the equation below: =1−−/ (2) Where e is 2. When the capacitor is discharging the same CR formula applies, as the capacitor also discharges in an exponential fashion, quickly at first and then more slowly. An audio crossover circuit consisting of three LC circuits, each tuned to a different natural frequency is shown to the right. When an inductor is charging, the energy is stored in the magnetic field and that stored energy is used to drive the current while discharging of the inductor. STUDY OF RC AND RL CIRCUITS Venue: Microelectronics Laboratory in E2 L2 I. The bigger τ is the longer it takes for the circuit energy to discharge. Clamp circuit is a circuit of the capacitor charging and discharging of the external input Cin. The simple circuit at the right demonstrates the discharging of a capacitor initially carrying charge Q0. a circuit’s time constant determines how it is affected by an RC circuit. Mutual repulsion of like charges in the capacitor progressively slows the flow as the capacitor is charged, stopping the current when the capacitor is fully charged and Q=C⋅emf. clear distinction between a resistor and inductor. Ninth Edition. In a practical clamping circuit, the values of C and RL are so chosen that discharging time is very large. 8 Efficiency 13. Personal Computers Have Become An Essential Part Of The Physics Curricula And Is Becoming An Increasingly Important Tool In The Training Of Students. In both the half cycles, the flow of current will be in the similar direction across the RL load resistor. enable you to estimate the performance of your capacitive discharge welder, 2. 3, Charging and Discharging a Capacitor, with PhET Simulation RC Circuits No. A RC Circuit consists of a Resistor and a Capacitor, RL circuit consists of Resistor and Inductor, and RLC circuit consists of a Resistor, Capacitor and Inductor. This is initially low, but rising as C1 charges. The formulas on this page are associated with a series RLC circuit discharge since this is the primary model for most high voltage and pulsed power discharge circuits. Book Description A comprehensive collection of 8 books in 1 offering electronics guidance that can't be found anywhere else! If you know a breadboard from a breadbox but want to take your hobby electronics skills to the next level, this is the only reference you need. Build and Simulate a Simple Circuit Introduction. What are the new voltages and charges? Charge inductors and discharge them through other inductors. C R! Charging Discharging Discharging For the discharging case, applying the loop rule to the circuit gives: Chapter H - Inductance and Transient Circuits 3. PHYS 142 FORMULA SHEET Nicholas Salloum Electromotive Force and Power EMF of ideal source: 𝜀= å â æ æ EMF of source with internal resistance: =𝜀−. , a coil with an inductance L) in series with a battery of emf ε and a resistor of resistance R. Once the voltage supply becomes superior to the voltage of the capacitor, the capacitor gets charging. Draw the other for discharging a capacitor through a resistor, as in the circuit in Figure 2 (above), starting at t = 0, with an initial charge Q o. Circuit prototyping breadboard resistor RL is placed in parallel with the capacitor. C + v - i Figure 1. Capacitors Initial and Final Response to a "Step Function" • Inductors and Capacitors react differently to a Voltage step • Just after the step Capacitors act as a short if uncharged dt dV IC(t )=C • If charged Capacitor acts as an voltage source • As time goes to infinity change in voltage goes to zero. Physics 102: Lecture 7, Slide 2 (even if only fractions of a second). Circuit Diagram Battery Charger Using Scr. 4, the output voltage has a very low amplitude because it barely has time to charge. Hint: the calculated values have to be made when the circuit is in AC steady-state*. The process of charging up a capacitor from a power supply takes time. Theoretically, the time constant is given by the product of the resistance and capacitance in the circuit, RC. After one time constant 𝜏𝜏, the current is 1/𝑒𝑒 of its initial value. 5 As RL varies from 0 to 12 in increments of 2\u2126, calculate the power dissipated by RL. The parallel RC circuit shown to the right behaves very differently when AC is applied to it, than when DC is applied. A stopwatch was started when the switch was thrown to start the charge/discharge, and the DMM read at known times until the voltage essentially stopped changing. Electron Motion During Charge. Capacitors are the electrical analog of springs. The difficulty arises in trying to determine which batteries are powering the circuit, or discharging, and which one are charging. txt) or view presentation slides online. For your example, that would be 3,300 ohms * 0. Ripple factor: Ripple factor is a measure of effectiveness of a rectifier circuit. Thus τ C of CR circuit is the time which the charge on capacitor grows from 0 to. The one thing that struck out the most ie the fastest charge up and discharge time was the circuit with the resistance of 100k and the capacitance of 47 uF. ABSTRACT A charging and discharging circuit is formed by a PNP tran-sistor which is turned on by current flow through an isolating diode and a base bias network only when the input potential becomes higher than the reference potential, thereby closing a first charging path through a first resistor and another isolating diode to a common capacitor; an NPN transistor which is turned on by. When a capacitor is charging, current flows from a voltage source through the capacitor. Other features of this circuit are a variable regulated DC voltage output (0-12V), voltage display panel meter, provision to measure charging/discharging current, ammeter, and micro soldering iron. Charge oF an RC Series Circuit Discharge Cycle. 1 RC Circuit Analysis No. The time constant is related to the cutoff frequency f c, an alternative parameter of the RC circuit, by = = or, equivalently, = = where resistance in ohms and capacitance in farads yields the time constant in seconds or the frequency in Hz. Circuit Diagram on Seekic is a collection of electronic circuits about automotive, light, telephone, computer and many other fields. The solution is then time-dependent: the current is a function of time. Find the natural oscillation frequency. A rst example Consider the following circuit, whose voltage source provides v in(t) = 0 for t<0, and v. The simple rules used for analyzing networks with only one battery and a collection of resistors do not work as well when you introduce one or more additional batteries. Although they all took different circuit approaches, the essence of each one resembled the overall topology shown above. 8 percent) of its initial voltage is known as the TIME CONSTANT (TC) of the circuit. Each of the following waveform plots can be clicked on to open up the full size graph in a separate window. R,expt (t) on the same graph paper. My problem is they keep going dead. The charge of the capacitor,the current in the circuit, and, correspondingly, the voltages across the resistor and the capacitor, will be changing. Figure 2: Discharging Capacitor. Once the magnetic field is up and no longer changing, the inductor acts like a short circuit. Some responses. This voltage is of the opposite polarity of the applied voltage. (3 Marks) Solution. We write equations for across the inductor. Before we solve a first order differential equation, let’s consider an example circuit. The time constant of a series RC circuit is defined as the time taken by the capacitor to charge up to 63. If the inductor is initially uncharged and we want to charge it by inserting a voltage source V s in the RL circuit: The inductor initially has a very high resistance, as energy is going into building up a magnetic field. This size of this resistance is such that the associated power loss is equal to the sum of the insulation and. Circuit prototyping breadboard resistor RL is placed in parallel with the capacitor. To get comfortable with this process, you simply need to practice applying it to different types of circuits such as an RC (resistor-capacitor) circuit, an RL (resistor-inductor) circuit, and an RLC (resistor-inductor-capacitor) circuit. Most students complete part A in week one and part B in week two. 5, Charging a Capacitor, Voltage and Current, Worked Example RC Circuits No. Consider a circuit consisting of Resistor (R Ohms) , Capacitor (C Farads) , a Voltage Source (V voltage) and a switch as shown below:. Current Current = rate of change of charge SI Unit: 1 ampere = 1 A = 1 C/s Current direction is defined as the direction of positive “charge carriers”, but actually. Example 2 - Charging / discharging RC circuit. Charging and discharging the RC circuit Charging Initially, a capacitor is in series with a resistor and disconnected from a battery so it is uncharged. t = 0 the switch is in position a and then a very long time later compared to the characteristic time constant of the circuit, the switch is flipped to position b. But when they supply energy to the circuit, the "voltage drop", technically a "voltage rise" or "negative voltage drop" flows out the + terminal and into the negative terminal, just like a voltage or current source. What are the new voltages and charges? Charge inductors and discharge them through other inductors. It covers Inductor charging and discharging in RL Circuit. Shunt Capacitor Filter and types of it: This is the most simple form of the filter circuit and in this arrangement a high value capacitor C is placed directly across the output terminals, as shown in figure. Please select whether you prefer to view the MDPI pages with a view tailored for mobile displays or to view the MDPI pages in the normal scrollable desktop version.

Current, resistance, and resistivity review (article) | Khan Academy Current, Resistance, Voltage, and Power. Current Current is a measure of the flow of electric charge through a material. A material that can The temperature- dependent resistivity ρ(T) can be found using the formula, AP Physics Notes. Mr. Andersen describes the relationship between voltage, current and resistance in an electric circuit. Ohm's Law is introduced through a circuit simulation. 1 point. Example: The full battery current passes through both A and D, so they have the same current. Because they have the same resistance, Δ. Δ. = A. D. V. Current I is the amount of charge per time that passes through an area perpendicular to the flow: Current is measured in SI units of amperes Aand This definition for current can be applied to charges moving in a wire, in an electrolytic cell, or even in ionized gases. In visualizing charge flowing through a circuit, it is not accurate to imagine the electrons moving very rapidly around the circuit. At this rate, the time to travel 10 cm is about 11 minutes. It is obvious from experience that it does not take this long for a bulb to glow after the switch is closed. When the circuit is completed, the entire charge distribution responds almost immediately to the electric field and is set in motion almost simultaneously, even though individual charges move slowly. The battery provides a voltage V between its terminals. The electric field set up in a wire connected to the battery terminals causes the current to flow, which occurs when the current has a complete conducting path from one terminal of the batter to the other—called a circuit. Introduction to circuits and Ohm's law (video) | Khan Academy By convention, the direction of current in the external circuit not in the battery is the direction of motion of positive charges. In metals, the electrons are the moving charges, so the definition of the direction of current is opposite the actual flow of the negative charges in a wire. Electric fields are not found in conductors with static charges as shown by Gauss's law, but electric fields can exist in a conductor when charges are in motion. The potential difference between the terminals of the battery when no current is present is called the electromotive force emf. The historical term emf is a misnomer because it is measured in volts, not force units, but the terminology is still commonly used. Resistance and resistivity Experimentally, it was found that current is proportional to voltage for conductors. The proportionality constant is the resistance in the circuit. So first, let me construct a battery. So this is my battery. And the convention is my negative terminal is the shorter line here. So I could say that's the negative terminal, that is the positive terminal. Associated with that battery, I could have some voltage. And just to make this tangible, let's say the voltage is equal to 16 volts across this battery. And so one way to think about it is the potential energy per unit charge, let's say we have electrons here at the negative terminal, the potential energy per coulomb here is 16 volts. These electrons, if they have a path, would go to the positive terminal. And so we can provide a path. Let me draw it like this. At first, I'm gonna not make the path available to the electrons, I'm gonna have an open circuit here. I'm gonna make this path for the electrons. And so as long as our circuit is open like this, this is actually analogous to the closed pipe. The electrons, there is no way for them to get to the positive terminal. But if we were to close the circuit right over here, if we were to close it, then all of a sudden, the electrons could begin to flow through this circuit in an analogous way to the way that the water would flow down this pipe. Now when you see a schematic diagram like this, when you just see these lines, those usually denote something that has no resistance. But that's very theoretical. In practice, even a very simple wire that's a good conductor would have some resistance. And the way that we denote resistance is with a jagged line. And so let me draw resistance here. So that is how we denote it in a circuit diagram. Now let's say the resistance here is eight ohms. So my question to you is, given the voltage and given the resistance, what will be the current through this circuit? What is the rate at which charge will flow past a point in this circuit? Pause this video and try to figure it out. Well, to answer that question, you just have to go to Ohm's law. We wanna solve for current, we know the voltage, we know the resistance. So the current in this example is going to be our voltage which is 16 volts, divided by our resistance which is eight ohms. And so this is going to be 16 divided by eight is equal to two and the units for our current, which is charge per unit time, coulombs per second, you could say two coulombs per second, or you could say amperes. And we can denote amperes with a capital A. We talked about these electrons flowing, and you're gonna have two coulombs worth of electrons flowing per second past any point on this circuit. And it's true at any point, same reason that we saw over here. Current, resistance, and resistivity review Even though it's wider up here and it's narrower here, because of this bottleneck, the same amount of water that flows through this part of the pipe in a second would have to be the same amount that flows through that part of the pipe in a second. And that's why for this circuit, for this very simple circuit, the current that you would measure at that point, this point, and this point, would all be the same. But there is a quirk. Pause this video and think about what do you think would be the direction for the current? Well, if you knew about electrons and what was going on, you would say, well, the electrons are flowing in this direction. And so for this electric current, I would say that it was flowing in, I would denote the current going like that. Well, it turns out that the convention we use is the opposite of that. And that's really a historical quirk. When Benjamin Franklin was first studying circuits, he did not know about electrons. They would be discovered roughly years later. He just knew that what he was labeling as charge, and he arbitrarily labeled positive and negative, he just knew they were opposites, he knew something like charge was flowing. And so, in his studies of electricity, he denoted current as going from the positive to the negative terminal.

When we talk about the production of any product, we always say that the interaction of different elements and processes allows the raw material to become the final product, which is destined to get into the hands of the final consumer. Production costs are costs associated with the production of these final products. These can include material costs, labor costs, costs associated with promoting a product to the market and selling it, along with other costs. In business, you will also come across marginal costs. Let us give a definition of marginal costs. First, let’s point out that margin is one of the important parts of basic economics. Economists will tell you that marginal means one more. So, instead of saying marginal cost, you can also say the cost of one more (or one less). Marginal costs by definition are additional business costs for each additional increment or next unit of production in excess of the current volume, i.e. marginal cost reflects the variations in costs that are a result of an increase or decrease in production by one unit. The marginal cost formula is very simple as you can see below. MC = (TC2 – TC1) / (Q2 – Q1) = ΔTC / ΔQ Based on the marginal cost formula, we can say that these costs can also be viewed as the ratio of the increase in total costs for the reporting period to the number of manufactured products, by which the output increased during this reporting period. Mathematically, this means that the total cost invested to obtain a certain number of units must be divided by the quantity of actual units produced so that it can be defined as a unit cost. The calculation process can be done in several steps: - Find the change in costThe costs a business incurs typically vary with a reduction or increase in the number of goods it makes or services it provides. Obviously, you will have a larger change in cost with more services provided or goods produced and vice versa. This change in costs occurs due to a change in variable costs. Obviously, even the fixed costs can increase or decrease if the business crosses a specific threshold, e.g. production facility would need to be expanded if the company decided to manufacture more than 200,000 units. Thus, your first step is to simply take the total costs a business has now and subtract from it total costs it expects to have or incurred) after it changes the level of its activity. - Calculate the change in quantityYour next step would be to determine whether the business is adjusting its production or provision of services and by how much. Just like you did the calculation in the previous step, you would take the quantity before the change (the old quantity) and subtract that from the quantity after the change (the new quantity). - Divide the resultsNow that you got your numbers ready for the last step, all you have to do is divide the number you got in the first step, or your change in cost, by the number you got in the second step, or your change in quantity. Keep in mind that the marginal cost per unit you calculated for the additional 50 units will most likely differ from the marginal cost per unit you would get for the additional 51 units. Accordingly, it would need to be recalculated for all the possible output options. To make this process more time-efficient, you can consider online calculators created specifically for this. Keeping proper financial records is time-intensive and small mistakes can be costly. BooksTime makes sure your numbers are 100% accurate so you can focus on growing your business. Purpose and analysis Why would management want or need to calculate the marginal cost in the first place? When thinking of margin, economists are thinking about the cost or benefit of making a decision, e.g. buying another piece of production equipment or making another batch of goods. As the quantity goes up, the marginal costs grow while the marginal benefit decreases. Simply put, the making of another piece brings additional costs and additional profits. As long as the marginal cost is below the market price level, the production is profitable. When it starts to exceed the price, it is an indicator of reduced efficiency. The marginal cost analysis is very useful in project implementation because, from a financial point of view, the optimal point is between the production cost and selling price, so that an appropriate price is calculated at which the company does not lose money and at the same time does not mistreat the customer. Undoubtedly, taking this into account when planning projects will help to get the best financial result. The curve representing marginal cost above depends only on the variable costs, while the effect of fixed costs is included along with the variable figures in the average cost. The marginal curve does not take into account fixed costs because they exist regardless of whether a business has more output or not. Initially, marginal costs are reduced and stay lower than the average. This is because if unit costs decrease, then each subsequent product is cheaper than the one made before it. The subsequent growth of marginal cost means that each following unit of production becomes more expensive. The marginal costs line crosses the average costs line at its minimum point M. Comparison and evaluation of various production costs is important information for the management of a company that is trying to determine the optimal level of business activity. At point M, the offer price coincides with the average and marginal costs. This point represents the equilibrium. When moving from point M to the right, an increase in production leads to a decrease in profit because, for each unit of goods, additional costs grow. Going beyond point M leads to instability of the company’s finances. It is possible to directly and immediately control the marginal cost as opposed to any other costs a business might have. In fact, this is what it costs the company to have the last item made for ready for the customer, and at the same time the costs that can be cut down by reducing the volume of production by this last unit. The figures of average cost do not provide this information. Output decisions are usually based on margins, that is, decisions about whether a business produces one unit more or one less. In combination with the marginal benefit indicator (reflects the change in income with an increase or decrease in production per unit of product), the marginal cost indicator allows management to determine the profitability of a particular change in the scale of production. Let us illustrate the marginal cost formula with an example. The total costs North Inc. incurred in June 2020 amounted to $62,500. The number of manufactured goods is 1,400 units. In July, additional 184 units were produced. Total costs for July 2020 amounted to $66,500. In other words, the total costs increased by $4,000 To calculate the marginal costs for North Inc., we will input the numbers we are given into the marginal cost formula presented earlier. MC = ($66,500 – 62,500) / (1,584 – 1,400) MC = $4,000/184 MC = $21.74 Based on the calculations we just did, we can say that each extra unit the company made in July cost North Inc. $21.74. It was a reasonable decision to produce more goods because the average cost per unit in June was $44.64, so the company is at a point where the benefits of making more goods significantly overweight the associated costs. Moreover, after further analysis, it might be likely that North Inc. should consider increasing its production even more to rip maximum benefits and make more profit. Author: Charles Lutwidge

It probably says more about me than anything else, but I've been asked the following question at least half-a-dozen times. In the interests of having a place to point future questioners I offer the following answer to the (apparently common) question: if the temperature today is 0°C, and it will be twice as cold tomorrow, what will the temperature be tomorrow? And the instant answer is: Which is, in human terms, rather cold. The lowest temperature recorded and confirmed on Earth is -89.4°C, recorded on 21st July 1983, at Vostok, a Russian research station in Antarctica. It's also a rather dramatic drop in temperature. A lot colder than most people guess when they ask this or similar questions. So, how did I arrive at this apparently drastic figure, and how do you divide zero by two and get a number other than zero in the first place? The answers to both questions derive from the same, usually overlooked, point: the Celsius temperature scale, like the Fahrenheit scale (and many other, now obsolete, temperature scales such as the Newton, Romer, Delisle, Leyden, Dalton, Wedgewood, Hales, Ducrest, Edinburgh and Florentine scales) is a relative scale. 0°C isn't the same as 0mm wide or 0V of electrical potential. Both these latter are absolute measures. You can't get narrower than 0mm and you can't get less electrical potential than 0V. You can get colder than 0°C, however, since 0°C is just the freezing point of water. So, to arrive at the frigid forecast above I simply converted the first figure to an absolute temperature scale (the Kelvin scale), halved it and then converted it back to Celsius. Makes sense now? I didn't think so. Let's step back a bit and take a look at the relative temperature scales, starting with the oldest temperature scale still in regular use. The Fahrenheit scale, developed in 1724 by Gabriel Fahrenheit, used mercury to measure changes in temperature, since mercury exhibits consistent changes when it undergoes thermal change. Mercury expands and contracts as the temperature changes and this volume change is both uniform across a wide range and large enough to measure accurately. In addition, mercury is cohesive rather than adhesive, so it doesn't stick to the only transparent substance Fahrenheit had access to: glass. Finally, mercury is bright silver, making it easy to visually distinguish changes in liquid volume in a narrow tube. Fahrenheit began by placing his mercury thermometer in a mixture of salt, ice and water. The point the mercury settled to on his thermometer was considered zero. He then placed the thermometer in a mixture of ice and water. The point the mercury settled to this time was set as 30. Finally, 'if the thermometer is placed in the mouth so as to acquire the heat of a healthy man' the point the mercury reaches is set to 96. Using this scale, water boils at 212 and it freezes at 32. This latter number is an adjusted figure on Fahrenheit's part: it made the difference between boiling and freezing a relatively clean 180. [NB, the above chronology isn't the only possible process Fahrenheit undertook. The Wikipedia article on the Fahrenheit temperature scale notes several other mooted explanations. Cecil Adams's The Straight Dope site also covers the origins of the Fahrenheit scale, focusing on the more amusing (or bemusing) possibilities.] Less than twenty years after Fahrenheit's scale was developed, the Celsius scale was created by Swedish astronomer Anders Celsius. His scale used the freezing and boiling points of water as the two key markers and put 100 degrees between the two temperatures. Unlike today, however, in Celsius's original scale, water's boiling point was 0 and the freezing point 100. In the years after his death in 1744, the numbering scheme was reversed. This change is routinely credited to another great Swede, Carl Linnaeus (also known as Carolus Linnaeus) but the evidence for this is circumstantial and not particularly convincing. Numbering scheme aside, the modern Celsius scale (used pretty much everywhere on earth except the United States) is different from the one Celsius developed. It doesn't make much difference in day-to-day use, but the basis of the modern Celsius scale is the triple-point of water. The triple-point of a substance is the temperature and pressure at which the solid, liquid and gaseous states of said substance can all co-exist in equilibrium. And the triple-point of water is defined as 0.01°C. As well, each degree Celsius is now defined abstractly. In Celsius's original scale a one degree change in temperature was defined as a 1% change in relative temperature between two externally referenced circumstances (ie the boiling and freezing points of water). Today, a degree Celsius is defined as the temperature change equivalent to a single degree change on the ideal gas scale. The ideal gas scale brings us almost to the point (finally, I hear you cry). As noted at the beginning of all this, the temperature scales above are relative scales: they give you a useful number to describe the thermodynamic energy of a system but they do so by creating a scale which is relative to some physical standard (whether that be the triple-point of water or the 'heat of a healthy man'). Back in 1787, however, Jacques Charles was able to prove that, for any given increase in temperature, T, all gases undergo an equivalent increase in volume, V. Rather handily, this allows us to predict gaseous behaviour without reference to the particular gas being examined. It's as if gases were fulfilling some Platonic conceit, all acting in a fashion essentially identical to an imagined ideal gas. Hence the 'ideal gas scale' which describes the behaviour of gases under changing pressure without reference to any particular gas. The Platonic ideal falls apart at very high pressures because of simple physical and chemical interactions. For the sort of pressures needed to use a gas as a thermometric medium (ie measurer of temperature) on earth, however, all gases exhibit the same, very simple behaviour described by the following equation: pV = [constant]T or in words: pressure multiplied by Volume = [a derived constant] multiplied by Temperature Which means if you keep the pressure constant, as the temperature changes so does the volume. Or, if you change the temperature and keep the volume constant, the pressure goes up or down in direct relation to the temperature's rise or fall. One very nifty thing about this is the way it makes it possible to create a temperature scale which is independent of the medium used to delineate the scale. Back in 1887, P Chappuis conducted studies using gas thermometers which used hydrogen, nitrogen, and carbon dioxide as the thermometric media at the International Bureau of Weights and Measures (BIPM). Regardless of the gas he used, he found very little difference in the temperature scale generated. If the temperature of the gases changed by a value T, and the pressure, P, was held still, the increase in volume, V, was the same regardless of the gas being used to set the scale. This change in thermodynamic activity has been recognised and accepted as the fundamental measure of temperature, since it is derived from measures of pressure and volume that aren't dependent on the substance being measured. One of the most important consequences of this discovery is the recognition that there is a naturally defined absolute zero temperature value. When the pressure exerted by a gas reaches zero, the temperature is also zero. It is impossible to get 'colder' than this, since at this temperature all atomic and sub-atomic activity has ceased. (And, before anyone asks, yes I know what negative temperature is, and it isn't a temperature 'below absolute zero.' Systems with negative temperature are actually hotter than they are when they have positive temperature.) In 1933 the International Committee of Weights and Measures adopted a scale system based on absolute temperature. It is called the Kelvin scale and uses the same unitary value for single degrees as the modern Celsius scale. So a one-degree change as measured by the Kelvin scale represents the same change in temperature as a one-degree change as measured using the Celsius scale. The zero-point for the Kelvin scale, however isn't an arbitrary one (eg the freezing point of water) but the absolute one. Absolute zero is, as it happens, equivalent to -273.15 C, so converting between K and C is a simple matter of addition or subtraction: C = K - 273.15 K = C + 273.15 So 0 degrees Celsius is 273.15 Kelvin. Using standard notation for each scale we can re-state this sentence thus: 0°C = 273.15K Note there is no degree symbol used when denoting a temperature in Kelvin. And, just as there is no degree symbol, the word isn't used either. The phrase 'degrees Kelvin' is incorrect: just use the word 'Kelvin.' Which brings us, finally, to explaining how I arrived at the temperature I listed at the beginning of this article. As I noted above, the Celsius and Fahrenheit scales are relative scales, so you can't compare two different temperatures measured using these scales absolutely. 20°C is not twice as warm as 10°C, since both are a measure relative to the triple point of water. The Kelvin scale, however, is an absolute scale. Different values measured using this scale are related in absolute comparative terms. 20K is twice as warm as 10K (although both values are pretty damned cold relative to what you or I are comfortable with). So, to find out what temperature (in degrees Celsius) would be 'twice as cold' (ie half the temperature) of 0°C I simply converted the value to Kelvin: 0 + 273.15 = 273.15K Divided this value by 2: 273.15/2 = 136.575 K and converted it back to degrees Celsius: 136.575 - 273.15 = -136.575°C Working in the other direction, twice as hot as 0°C is easy to calculate. It's 273.15°C. Which is rather hotter than any human can handle. If nothing else, this demonstrates how narrow a range of temperatures suit human beings. Let's presume -10°C – 50°C is a useful range of liveable temperatures for human beings. I'm being generous with this range. The low is, in human terms, well below the freezing point of water. And the high is, again in human terms, a long way above blood temperature. This range is only acceptable as a liveable range if we assume 1) the range refers to measured temperatures and 2) we have technology capable of keeping experienced temperatures (eg, in a dwelling or next to human skin) from reaching these extremes of heat and cold. Converting this to Kelvin, we have a range of 263K – 323K. (I'm leaving the 0.15 off: it doesn't change the arithmetic, other than to needlessly complicate things.) The lowest temperature in this range is 81% of the highest temperature in this range. 323K (50°C) is only 19% warmer than 263K (-10°C). Change the liveable range to 0°C – 40°C (a range more genuinely liveable, especially if we assume only basic available technology) and the hottest we can reasonably handle is only 13% warmer than the coldest we can live with. Be even more conservative, and restrict the range so it runs roughly through the human comfort zone: 10°C – 35°C (a range that goes from 'cold if you don't have warm clothing' through to 'hot in the sun but bearable if there's almost any sort of breeze') and the hottest weather we can comfortably manage is only 9% warmer than the coldest most of us are willing to deal with. No wonder folk are concerned about a 0.6°C increase in global surface temperatures over the last 100 years.

My Theory of Attention is based upon a Math/Fact Matrix – the Living Matrix, to be precise. To facilitate a common understanding, let us provide some definitions. A Math/Fact Matrix is a complex web of interlocking relationships between a mathematical system and scientific facts. The Material Matrix is the most famous, as it is at the heart of our technological wonderland. Newtonian Physics provides the mathematics while material behavior provides the evidence. The Living Matrix is another type of Math/Fact Matrix. The Physics of Information provides the mathematical system, while evidence surrounding Attention provide the scientific facts. Another intent of this volume is to establish the validity of the Living Matrix. Let’s define the components of our Math/Fact Matrix. Fact, in this limited context, only refers to scientific facts. We use the term broadly to indicate any kind of belief that is widely held by scientists in any field. There must be virtually universal acceptance of the belief’s validity amongst the scientific community. The certainty must be so strong that the belief is generally taken as a fact. In other words, scientific facts are accepted as unquestionably true. Scientific facts can be take many forms: 1) a simple measurement, e.g. that board is 2” by 4”, 2) simple experimental findings, e.g. a water molecule consists of 2 hydrogen atoms and 1 oxygen atom; 3) complex living behaviors, e.g. sleep deprivation harms cognitive performance, or 4) a complex theory, e.g. Newton’s law of gravity and Darwin’s theory of evolution. A general scientific fact that is pertinent to our study: evolutionary forces determine biological forms. These facts generally define a phenomenon or its interactions with other phenomenon. The phenomenon can be specific or general. Humans have two eyes; living systems are composed of cells; cells cooperate as a group to create organisms. Reiterating for retention, the Physics of Attention derived from a Math/Fact Matrix. The facts in this matrix are not just numerical measurements, but instead beliefs that are universally held by the scientific community – Scientific Facts. What about the Math part of the matrix? The mathematics of a Math/Fact Matrix is a system. In order to be considered part of a matrix, this system must coincide with a set of scientific facts. This type of mathematical system can take many forms. The most spectacular is Newtonian Dynamics, which provides the dynamical structure of the so-called Material Matrix. Calculus provides the mathematical structure of the system. The interaction between the math system and material evidence gave rise to the Material Matrix, i.e. the Physics of Matter. Data Stream Dynamics (DSD) provides the physics, i.e. the dynamical structure, for our particular matrix – the so-called Living Matrix. The Living Algorithm (LA) provides the mathematical structure for DSD. The interaction between the math system and living evidence gave rise to the Living Matrix, i.e. the Physics of Information. The Physics of Matter and the Physics of Information are entirely different in that they belong to mutually exclusive sets. Calculus is based in Closed and Regular Equations. In other words, these equations are closed to external input and don’t refer to themselves. In contrast, the Living Algorithm is an Open and Reflexive Equation. Put another way, the LA is open to external input and past results play a part in determining current results. The Living Matrix gave birth to my Theory of Attention. The Material Matrix gave birth to the Theory of Matter. The Theory of Matter is so well-established that it could be considered a scientific fact. On the other hand, my Theory of Attention has a fan base of one. It is for this reason that I have written so many supportive pages. A Math/Fact Matrix indicates that there are distinct patterns of correspondence between a mathematical system and scientific facts. How is the correspondence validated or verified? According to my Theory of Attention, there are at least 3 mathematical realms of existence. Each realm has a different relationship with the facts. Let us explore the nature of these relationships one realm at a time. 1) Molecular Realm: Scientists access this realm through direct observation. They first accumulate precise measurements regarding the absolute essences of particles/objects as they move through space. These measurements are rendered in numbers. For instance, an iron ball weighs 2 pounds and is 4 inches in diameter. Equations are then written to match the numerical patterns. Validated formulas become scientific facts, e.g. E = mc2. In this realm, objects, even atoms, can be observed moving through space and time. 2) Subatomic Realm: In contrast, scientists can only infer the existence of this realm through indirect observation. Because Subatomic entities, such as electrons and photons, are not observable, scientists deduce their existence by their effects. For instance, sound or visual trails are examined to ascertain the characteristics of the bizarre inhabitants of this peculiar realm. This data is also numerical. 3) Living Realm of Attention: Like the other realms of existence, the Math/Fact matrix is based upon patterns of correspondence between phenomena that are established scientific facts and a mathematical system. However, the inferences regarding the characteristics of this realm are generally based upon visually based evidence. Rather than numerical data, we employ visual pattern recognition in order to establish the correspondences. Put another way, an examination of the mathematical behavior of LA’s model reveals similarities with living behavior. It is evident that the Math-Fact Matrix associated with Attention lacks the empirical rigor of the Material Matrices. Despite this deficiency, three factors establish the validity of the Attention Matrix: 1) It provides explanatory power for a vast range of phenomena that remain scientific mysteries. 2) The Math and Fact network are isometric systems in that they share a common underlying inferential structure, i.e. exhibit a high degree of logical symmetry, and 3) redundancy logic, i.e. cross-validation from multiple sources. These same factors validate each Math-Fact Matrix. Let us provide a brief introduction to each factor in the so-called validation network for the Attention Matrix starting with explanatory power. Our model provides explanatory power for a wide range of phenomena from a variety of disciplines. Some of these experimentally verified phenomena remain scientific mysteries despite intensive research and speculation. The community understands ‘What’ happens, but have yet to uncover the ‘Why’ behind these enigmatic phenomena. Conversely, our mathematical model provides plausible explanations that link countless phenomena from a multiple disciplines under one roof. What does the roof consist of? And what binds it together? Isometric logic: the process of employing the implicit logical structure of a known system to better understand the implicit logical structure of an unknown system. Humans use this process to form verbal abstractions; scientists use this process as a powerful tool. They employ the implicit logic of mathematical systems to better understand of implicit logic of empirical systems. As another form of validation, the LA system has an isomorphic relationship with a network of phenomena associated with Attention. Redundancy Logic is the final factor validating the Math-Fact Matrix that defines the Realm of Attention. According to this extraordinary type of logic: the greater the number of sources pointing to the same conclusion, the more likely the conclusion is true. Living systems employ this form of logic to affirm a common reality. Scientists employ Redundancy Logic as a means of forming the common consensus that is the foundation of validating scientific theory. 1) There are an uncountable number of patterns of correspondences between scientific facts and the LA’s mathematical system. 2) These scientific facts span a wide range of phenomena from a variety of disciplines. 3) DSD provides explanatory power for many scientific mysteries. 4) DSD and a phenomenal system have an isometric relationship, i.e. a high degree of logical symmetry. Each of these sources support the conclusion that DSD provides a good model for the network of phenomena associated with the Realm of Attention. The redundancy of these multiple forms of evidence provides overwhelming support for our model. Finally, our Theory of Attention matches and extends our common sense notions regarding intention, mental energy, and choice. With this addition, there are three forms of validation for the conclusions behind the Theory of Attention: 1) cross-disciplinary evidence linked under one roof, 2) explanatory power for scientific mysteries, and 3) congruence with our common sense regarding choice. This three part logical redundancy generates a virtually indestructible logical web of understanding. What is the significance of my work? Attention Theory provides a theoretical foundation for our common sense understanding of choice. The mathematically based theory integrates intention, information, attention, mental energy and experience in an elegant package that embodies simplicity. Its significance lies in its ability to extend our understanding of Attention. With deeper insights into this fundamental feature of living experience, we can maximize the quality of our lives. With this knowledge, we can more easily both fulfill potentials and avoid pitfalls. As mentioned, the Living Matrix provides the foundation for my Theory of Attention. Due to its foundational importance, this entire work is devoted to establishing the validity of the matrix. To that end, we examine the myriad patterns of correspondence between living behavior and the mathematical behavior of our model. More specifically, we examine the parallels between well-established Attention and Sleep related phenomena and the behavior of the two primary forms of our mathematical system, the Pulse and the Triple Pulse. As will be exhibited, there are widespread patterns of correspondence between the mathematical behavior of these two forms and many aspects of living behavior, especially those regarding Attention. You are probably wondering, how could mathematics possibly apply to imprecise, unpredictable living behavior? If it does apply to behavior, what are the implications? Could an understanding of the dynamics of behavior actually improve mental performance and the overall quality of life? Could the employment of these mathematical mechanisms in our day-to-day life actually improve our chances of having an 'optimal experience' - moving into the ‘Zone’ – becoming ‘one with the Tao’?

Physics Original Theory Research Series 1 2020/12/26 S.Asada Corrections and additions are made at any time →Japanese edition日本語版 It is an observational fact that dark matter, an unknown substance that is not observed but has the effect of creating a gravitational field, exists in large quantities in this universe, and that the total amount of it exceeds the total amount of visible stars and galaxies. . In this paper, we will elucidate the identity of this dark matter with our own theory, and clarify how it is made and what kind of properties it has. 0. Summary, Conclusion 0-1. What creates a gravitational field is matter itself, not "mass or energy" For the proof of this, see " 2. Proof of existence of vanishing mass". It may be slightly different from what the Einstein equation indicates, but according to this theory, inertial mass and its equivalent energy do not have the ability to create a gravitational field ( a field that stretches the time axis of space and slows down the speed of time). We usually call the property of receiving acceleration or generating inertial force in the gravitational field "inertial mass = gravitational mass" or simply " mass". Furthermore, mass and energy are equivalent in this property (E=Mc^2). In other words, energy has the properties of inertial mass and gravitational mass. (However, energy alone, such as a photon, does not show the property of mass, and it shows its property only when it is confined in a substance → See "The True Nature of Gravity ") In the conventional standard theory, the gravitational field is a distortion of space-time, and its magnitude is "described by the energy and momentum tensors". Roughly speaking , this means that it is mainly mass energy that creates the gravitational field. In other words, it is the claim that "mass receives acceleration in the gravitational field and also creates the gravitational field." However, according to this theory, the cause of creating a gravitational field is due to another kind of ability that matter itself has, and inertial mass and energy do not have such ability. The property of inertial mass is only passive. Matter is the only entity whose four-dimensional volume (x, y, z, t) is preserved. This is very different from the energy that the total amount is conserved in three-dimensional space. In a three-dimensional space, a substance can change its volume (x, y, z) as the time axis changes. As the time axis (t-axis) extends, the three-dimensional volume decreases in inverse proportion. But the four-dimensional volume does not change. The ability to create a gravitational field is proportional to the four-dimensional volume of matter. Specifically, the three-dimensional volume (=inertial mass) divided by the time velocity ratio with the observer is the amount of the property that creates the gravitational field (provisionally called "true mass"). In addition, matter acts on the time axis of the surrounding space in an amount proportional to its four-dimensional volume, extending it. In other words, time is delayed. This time delay in space is the " identity of the gravitational field ". The property of inertial mass is usually associated with matter. This is because matter contains energy proportional to its three-dimensional volume. Energy is the inertial mass itself. Inertial mass increases and decreases in the state of the time axis, and it is possible that some matter has completely disappeared. Even in such a case, the four-dimensional volume of matter does not change, so the ability to create a gravitational field is constant and does not change. 0-2. Phenomenon where matter loses mass As a phenomenon that is widely recognized as a general theorem (the law of conservation of mass and energy), when substances are bound by gravity or electric force, they always release binding energy. Then, the mass conversion value (ΔM=Ef/c^2) of the released binding energy is reduced from the total mass of the original substance. And when the bond energy becomes larger and larger, the larger bond energy associated with it is released. Then, it would be reasonable to think that the mass of matter may eventually become zero. We can witness that moment. For example, the combination (annihilation) of an electron and a positron. Details of this will be described later. 0-3. Matter exists even if it loses mass, and the gravitational field it creates does not change Even if the binding energy is released to the limit and the volume and mass become zero in the three-dimensional space, the matter continues to exist in the four-dimensional space-time. Mass is one of the properties of matter, but there are also states where it becomes zero. Just because mass is zero doesn't mean matter is gone. And the matter continues to retain the ability to create a gravitational field. In the pair annihilation described above, an electron-positron combination with no volume or mass is generated, but the gravitational field created by these does not change before and after the combination. Gravitational bonding and electrical bonding can certainly occur with bonding energies so huge that the mass becomes zero. Matter that has lost its mass in this way disappears, leaving only the property of creating a gravitational field, and becomes almost unobservable. My argument in this paper is that this is the identity of dark matter. 0-4. Properties and types of dark matter Specific dark matter candidates include electron-positron conjugates and proton -antiproton conjugates. These have zero volume and mass in three-dimensional space, as explained in detail in " Proposal of Four-Dimensional Volume Conservation Law " . For the proof of this, see "2. Proof of existence of vanishing mass" . Since they have no inertial mass, they probably move at the speed of light, and since they have zero volume in three-dimensional space, they usually interact with matter, pass through without colliding, and continue moving at the speed of light. The probability of existence will be widely spread. However, it has the ability to create a gravitational field as an invariant. In other words, it continues to maintain the gravitational field it had before the mass disappearance occurred. Such microscopic dark matter is so small that individual gravitational fields cannot be observed. However, since the number of existence is enormous, it will have a great influence . The above is mass loss due to coupling by electric force, but mass loss due to gravitational coupling between stars is also possible. However, since the masses of two stars are never exactly equal, they will not be completely massless after merging. In addition, when two stars are gravitationally coupled, they become a black hole before they can fully release their binding energy. In that case, the binding energy is also confined in the black hole, so that no further binding energy can be released. However, a black hole that has lost a large amount of mass due to such gravitational coupling has mass, but has a larger gravitational field than its mass. In other words, even though it has a small mass, it has a much larger gravitational field, so it may be treated as a kind of dark matter. 0-5. Generation and annihilation of electron-positron coupled dark matter The figure below shows the formation of dark matter by the combination of electrons and positrons. It also shows that dark matter can reseparate into electrons and positrons by absorbing high-energy photons. However, dark matter cannot absorb photon energy directly, and this reaction (pair production) occurs stochastically, such as when it collides with another nucleus and enters a high-energy state. The figure below shows the formation and disappearance of dark matter. When electrons and positrons combine with their strong electric force, they emit energy equal to their total mass in the form of photons, and themselves become entities with zero mass (vanishing mass dark matter). It has only the effect of creating a gravitational field and wanders through space. And when it absorbs a high-energy photon, it breaks the bond and splits again into an electron and a positron. Bonding is called "pair annihilation" and separation is called "pair creation". But if the theory is correct, the term may not be very correct. Sounds like a perfect theory, doesn't it? However, the illustration of the movement direction of the particles in the upper figure is not strictly correct. Actually, the law of conservation of momentum is obeyed. By the way, the electron-positron reaction also satisfies both the law of conservation of mass energy and the law of conservation of electric charge, and the gravitational field is also conserved. The mass of electron-positron combination = (electron, positron mass) 2 x 0.51Mev - (emission photon mass) 2 x 0.51Mev = 0, that is, the combination of electron and positron has zero mass. However, it retains the ability to create a gravitational field. This is eｰe⁺ coupled dark matter. There are also dark matter such as p-p‾ bonded type and ｎ-ｎ‾bonded type. 0-6. Behavior of dark matter and its effect on galactic motion When dark matter passes near or through the nuclei of ordinary matter, it is subject to some strong electric forces, temporarily loosening the bonds. If a high-energy photon acts at this moment, it may be absorbed and the bond may be broken to generate a pair. This is the production of positive and antimatter from dark matter. The motion of dark matter will change course due to the influence of electric force, etc., and it will become a zigzag motion. Due to the zigzag motion , dark matter is bound for a long period of time in areas with high celestial density such as galaxies. In addition, even if normal matter bends the course of dark matter by electric force, dark matter has zero mass energy, so it does not affect the behavior of ordinary matter. However , this is a two-party problem, and the situation is different in a large number of mixed states . Furthermore, the dark matter group that affects the behavior of ordinary matter receives a reaction from ordinary matter. This action also causes dark matter to be restrained to some extent by galaxies. In this way, dark matter is bound to the galaxy for a long period of time, and is distributed widely and in high concentration, especially in the outer part of the galaxy. And the gravitational field caused by these can explain the uniform rotation of the galaxy's outer periphery. In the region near the center of the galaxy, dark matter is trapped by the supermassive black hole at the core, so the dark matter density is lower than in the periphery. The supermassive black hole at the center of the galaxy grew rapidly, mainly by absorbing dark matter. There are restrictions on how black holes can absorb normal matter, making it difficult for them to grow rapidly. A black hole will have a large absorption cross-section for dark matter due to its spatial structure. A black hole that absorbs a lot of dark matter should have a small inertial mass compared to its gravitational force. Proof? In that case, it will be a long sentence, and it is unlikely that you will read it, so I presented the conclusion first. I wonder if it's okay~ The behavior of vanishing masses that have completely lost their mass, which I believe to be the identity of dark matter, is interesting. Its proper time has stopped, and its volume, inertial mass, and gravitational mass in three-dimensional space are also zero. Therefore, it travels at the speed of light and usually passes through matter without colliding with it. It is also unaffected by the gravitational field. However, it only has the effect of distorting space (to be precise, acting on the time axis to slow down the temporal velocity of space) and creating a gravitational field. → Detailed examination of the behavior of dark matter I try to prove the existence of such an object by a thought experiment. These proofs are possible within the realm of classical physics (mainly relativity theory), and do not require difficult and strange theories. It's not that you don't need it, it's that the owner can't use it I recognize that this theoretical development is slightly different from the common sense of modern physics, but I can't find any mistakes in my knowledge. If there are any mistakes, I would like to know, so I am waiting for the criticism of those who are more knowledgeable. Thank you . Please explain in such a simple way that even a junior high school student can understand it. I have already touched on the vanishing mass in The Story of the Birth of the Universe (My Original), but there was not enough depth to consider it as part of the whole . Only this will be examined in depth here. 2. Proof of existence of dark matter The main theme here is to prove that matter that has lost its mass and that still has a gravitational field (dark matter) is real. In particular , we clarify that the property of inertial mass and energy is different from the property and ability to create a gravitational field . As a general explanation of the gravitational field , the Einstein equation states that the source of the gravitational field (distortion of space-time) is the energy- momentum tensor consisting of energy and momentum (see wikipedia ), but this paper denies this. do. The reason for this is explained in detail by a thought experiment in the text. When multiple objects release binding energy -Ef by binding with gravity, electromagnetic force, nuclear force, etc., the mass corresponding to the binding energy Ef (ΔM = Ef/c ^ 2) disappears from the total of the original masses. is a well-known fact and there is no room for doubt. In order to expand the interpretation of this well-known phenomenon and to verify the results, we will tackle the following themes (1) to (3). Thereby, the above object is achieved. Assuming a state in which the bond energy is extremely high in the bonding of multiple objects, it is easy to imagine a situation in which the mass disappears from the substance due to the release of the bond energy. Note that the term “bond” is not limited to coalescence, but also refers to a state in which a bonding force acts in close proximity. And even if the object (matter) emits binding energy and the mass and volume in the three-dimensional space become zero, the substance continues to exist in the four-dimensional space-time. The volume and mass in the three-dimensional space are zero because the time axis has grown to infinity → Proposal of the four-dimensional volume conservation law In addition, mass is equivalent to energy and can completely escape from matter, but matter has the property of creating a gravitational field, which is immutable and cannot be separated. This property is called gravitational elementary weight here. I think that the invariants of the universe include mass energy, electric charge, and gravitational weight. 2-1. Proof of existence of dark matter The mass referred to here (inertial mass or gravitational mass) is a passive property such as receiving inertial force or receiving acceleration in a gravitational field. This does not include the active nature of creating a gravitational field. For example, the gravitational acceleration near the surface of the earth is 9.8m/s ^ 2, so a force of Mg=9.8N acts on an object with a mass of 1kg. Mass is equivalent to energy (the famous conversion formula is E=Mc ^ 2), and energy has the same properties as mass. In other words, it receives acceleration in the gravitational field and has inertia. However, I argue that mass and energy have no effect on creating a gravitational field. 2-1-1. Difference from conventional definition In the conventional definition, the gravitational field was also defined as "mass energy, momentum tensor", but here, "the property of receiving acceleration due to inertial force and the gravitational field" and "the property of creating the gravitational field " are treated as separate things. I needed it. Therefore, they are divided into two components: mass energy and true mass (gravitational elementary mass). A gravitational field is a distortion in space that has the ability to create it. This paper assumes that the ability to create a gravitational field is proportional to the four-dimensional volume of matter. Assuming that this four-dimensional volume is MT, MT can be substituted for mass M in the conventional gravitational equation. Objects passing through it move according to the distorted space regardless of the cause object that made it. In other words, it is not a force that simply acts between objects like nuclear force or electromagnetic force. 2-1-2. Adoption of the term "true mass = gravitational elementary weight" he ability to distort space and create gravitational fields is not associated with inertial mass, energy, or momentum, but rather as an independent property and associated with matter. As a representation of this property, the word "true mass = gravitational elementary weight" was applied here. 2-2. Explanation of the "thought experiment device" It is assumed that the thought experiment device in [Figure 1] is not affected by external forces such as gravity. Suppose that A and B are massive objects with completely equal masses. A gravitational force acts between A and B. However, A and B are connected by wires to the power generation motor, and the distance cannot be changed unless this rotates. The generator motor referred to here is an ideal converter that reversibly converts rotational energy into electrical energy, and the conversion efficiency is assumed to be 100%. Operate the generator motor so that A and B approach. The combined energy is then released and the generator motor generates electricity to charge the battery with electrical energy. Conversely, to separate A and B, electric energy can be supplied from the battery to work as a motor, and A and B can be separated. For simplicity, we assume that these movements are slow and the momentum is negligible. The direction of movement, speed, etc. can be controlled in any way by controlling the amount of energy supplied and released to the generator motor. 2-3. Case where mass energy is completely lost from matter → proof ① Assuming a state in which the bond energy is extremely high in the bonding of multiple objects, it is easy to assume that the mass will disappear from the substance. Note that the term “bond” is not limited to coalescence, but also refers to a state in which a bonding force acts in close proximity. For example, suppose two substances, A and B, are combined in a gravitational field. It is a well-known fact that then, the binding energy ΔEf is released and the mass (ΔM=ΔEf /c ^ 2) corresponding to the energy is reduced from the mass before binding. → Einstein's famous conversion formula E=Mc ^ 2 This phenomenon applies not only to gravitational coupling but also to electric force, magnetic force, chemical bond, nuclear force, and so on. This is explained using the thought experiment device in [Figure 1]. For example, since objects A and B have a large mass, they create a large gravitational field and exert a strong gravitational force. A generator motor stops this through a wire. Here, the generator motor is rotated slowly to shorten the distance between A and B, thereby releasing the binding energy and generating electricity, which is stored in the battery. Then, the mass of A+B decreases by the mass corresponding to the energy transferred to the battery (ΔM=Ef/c ^ 2). Conversely, the mass of the battery is strictly increased by (ΔM=Ef/c ^ 2). This keeps the mass-energy in box ② constant and satisfies the conservation law. Here, if the bond energy is extremely large, it is assumed that the mass of A and B becomes completely zero due to the release of large bond energy. It is easily assumed that such extremely large binding energy can be released in the case of binding by gravity and binding by electric force. Mass loss due to gravity is a phenomenon that occurs on a large scale, and mass loss due to electric force is a phenomenon that occurs in the microscopic region. In the case of the micro region, strong and weak forces other than the electric force will also participate in the bonding. 2-3-1. Mass loss due to gravitational coupling The binding energy due to gravity has the property that the binding energy increases infinitely as the total mass increases. So, for example, if A and B are neutron stars in the condition just before becoming a black hole (both are exactly the same in mass and other things), by bringing them closer, the bond energy release of both reaches the energy conversion value of their masses. , can have zero mass. In another example, if a large number of Earth-sized stars are brought close to each other at the same time, there may be a condition where the total binding energy reaches the total mass equivalent value even if they are not in contact with each other. Furthermore, it is known that the entire universe is very low density, but because it is vast, the total mass becomes huge, and the total amount of gravitational binding energy as a whole and the energy conversion value of the total matter are close. → gravitational binding energy For some reason, the condition where the mass becomes zero is close to or equal to the condition where it becomes a black hole. Also, complete mass disappearance due to gravity is unlikely in the natural state. Because if the masses of A and B are not exactly equal, the mass after binding will not be zero. In the case of gravitational coupling, it becomes a black hole in the process of gravitational coupling, and light energy cannot escape. Therefore, the mass energy is trapped and remains. So the loss would be about 50% of the total mass. 2-3-2. Mass loss due to electric force coupling Electrons and positrons, protons and antiprotons, neutrons and antineutrons, etc. also have a large electrical binding energy when the distance between their charges becomes extremely short, and it is assumed that the mass becomes zero when this is emitted. In this case, since the absolute values of the mass and charge of both are exactly equal, the mass after binding can be exactly zero. → binding energy Binding energy between charges Ef≒9×10^9×Q1×Q2 ÷r^2 (r is the distance between charges Q1-Q2)- - - - This has the same form as the gravitational binding energy formula According to a simple calculation, the distance between charges at which the mass becomes zero after this combination is about 10 ^ -15m in the case of an electron and a positron . In the case of protons, antiprotons, neutrons, and antineutrons, it seems a little complicated because they have an internal structure, but there is a distance between charges where the mass becomes zero. In the case of neutrons and antineutrons, the internal structure (quark level) has an electric charge similar to that of protons, and it is possible that they lose mass due to electrical coupling. → Confirmed in practice → (Example) electron-positron reaction, proton-antiproton reaction In the case of protons and antiprotons, it is not known whether they are combined in their original form or dispersed as quarks and anti-quark combinations. However, since pairing occurs, there is a high possibility that the mass disappears by combining in the form of protons and antiprotons, rather than by disjointed quark units. Furthermore, by combining unstable quarks, antiquarks, etc., it becomes a vanishing mass with zero mass, and by stopping the proper time, it can become a permanent life. Considering the case of electric coupling in the above [Fig. 1], the attractive force F becomes an electric force. Since the extracted energy is equivalent to mass, it has the properties of mass (inertial mass, gravitational mass), but does not have the property of creating a gravitational field. Objects A and B have the property of creating a gravitational field. 2-4. Matter and the gravitational field continue to exist even if the mass becomes zero → Proof 2 Matter continues to exist in four-dimensional space-time even if it releases its binding energy and becomes zero in mass and volume. The volume and mass in the three-dimensional space are zero because the time axis has grown to infinity → Proposal of the four-dimensional volume conservation law It is not mass or energy that creates the gravitational field . It is matter, and the ability of matter to create a gravitational field is proportional to its volume in four-dimensional spacetime. By the way, a gravitational field is a state in which the time axis of space is extended (time velocity is delayed) due to the "true mass = gravitational elementary amount" of a substance. In the previous proof, I came to the conclusion that the combined mass of substances A and B can become zero, but even if the combined mass of A and B becomes zero, A and B still exist. It's not that I stopped doing it. Even if the three-dimensional volume (mass) of matter becomes zero, the four-dimensional volume is preserved and the gravitational field does not change. We prove it below. 2-4-1. Reversibility of Mass Disappearance Phenomena Even if A and B release the binding energy in the thought experiment device of Figure 1 and the mass decreases infinitely, this process is reversible. If electric energy is supplied from the battery and the generator motor winds up the wires and separates A and B, the original masses of A and B will be restored. In order to explain the essence of this phenomenon in an easy-to-understand manner, let us assume that the objects are Mr. A and Mr. B, human beings with individuality and complex and delicate structures. When Mr. A and Mr. B are gravitationally coupled, gravitational coupling energy is released. Then, the mass of the two people decreases by the energy released, and the energy is converted into electrical energy by the generator through the wire and stored in the battery. The mass of the battery increases by that amount and satisfies the law of conservation of mass and energy in box ②. For the validity of this law of conservation, see " 3. Consideration of the Law of Conservation of Inertial Mass and Gravitational Field" at the end of this article. At first glance, it seems that Mr. A and Mr. B have moved to the battery, but since it is reversible, if you supply electricity to the generator motor and wind it up, it will return to the original Mr. A and Mr. B. Humans cannot easily be assembled from electrical energy, so we must think carefully about the meaning of reversibility. Even if the mass decreases or disappears, the matter of Mr. A and Mr. B always existed in Box ①. In other words, there is no change in the coordinates of Mr. A and Mr. B. Their mass does not change from their point of view. It is a mass transfer and disappearance phenomenon from the perspective of an external observer. Even if you think in terms of physical common sense, it is impossible for the substances that make up Mr. A and Mr. B to escape from Box ① through the rope. Only energy can escape through the rope. Also suppose that the energy is transferred to the battery and the mass in box ① approaches zero. If the gravitational field is caused by mass, then the gravitational field in box 1 will also approach zero. If that happens, even if you try to pull Mr. A and Mr. B apart, the gravitational force will be close to zero, so the energy from the battery will hardly flow back. In other words, the mass of Mr. A and Mr. B will not be recovered, and the reversibility is lost. This is physically impossible. Therefore, the gravitational field is not brought about by energy, but is created by the matter of Mr. A and Mr. B. Energy does not create a gravitational field. However, energy is equivalent to mass, has inertial mass, and receives acceleration in the gravitational field. 2-4-2. Proposal of four-dimensional volume conservation law to explain mass disappearance The phenomenon that occurs to Mr. A and Mr. B can be explained by the shape change in the four-dimensional space-time. Let x be the length on the X axis, y be the length on the Y axis, z be the length on the Z axis, and t be the length on the T axis (time axis) . It is reasonable to think that x, y , z , and t are conserved. When a large amount of bond energy is released by bonding, x, y , and z (= mass) are reduced so as to satisfy the conservation law in three-dimensional space , and energy is generated accordingly. In other words, matter is an entity whose four-dimensional volume is conserved, and energy is an entity whose total amount is conserved in a three-dimensional space. Since matter can be deformed in the direction of the time axis, its volume (=mass) in three-dimensional space can change, but energy is a quantity whose total amount is conserved in three-dimensional space, and cannot escape in the direction of time axis. Then, as x, y , and z are reduced so as to satisfy the conservation law in four-dimensional space-time, the time axis t is extended and the four-dimensional volume becomes constant, satisfying the conservation law. This means that the time of the substance is delayed. To explain this with humans, when a large amount of binding energy is released, the x, y , and z seen by an external observer, that is, the volume and mass, are observed to decrease, and t becomes longer, in other words, the lifetime is observed to become longer. . Ultimately, the volume and mass become zero, the time axis extends infinitely, and the proper time stops. 2-4-3. Mass loss and time delay appearing in all binding energy emission phenomena It is well known that time is delayed when binding energy is released in a gravitational field and mass is reduced, but according to this paper, this is not limited to the gravitational field, but all binding energies such as electric force, nuclear force, etc. I argue that the disappearance of mass and the time dilation occur on set in the event of the emission of . → It follows from the four-dimensional volume conservation law that I advocate. 2-5. Mass accompanies energy, gravitational field accompanies matter → proof ③ Mass is equivalent to energy and can be completely removed from matter, but matter has the property of creating a gravitational field, which is immutable and inseparable. This property is called "true mass = gravitational elementary weight" here. I think that the invariants of the universe include mass energy, electric charge, and gravitational weight. 2-5-1. Proposed law of conservation of mass and gravitational field Let us now consider the situation in Box ① in Figure 1. From here, energy can go in and out with wires, so there is no problem with the mass and energy in Box ① changing. However, since nothing goes in and out of box 2, there should be no change in black box 2 as seen from the outside no matter what happens inside (conservation law should hold). This is the mass, the gravitational field, and the electric charge. However, although the electric charge is preserved, there are positive and negative electric charges, so if the distance between the two is extremely close, the electric field will not reach the outside. See the extra edition at the end of the article for the proof of the correctness of this conservation law. 2-5-2. Whereabouts of the mass lost by the combination of A and B and whereabouts of the gravitational field The discussion proceeds on the premise that the above conservation law holds. Consider the phenomenon that when A and B approach each other, the bond energy is released and the mass is reduced. The reduced mass leaves Box ① in the form of mechanical energy through wires, is converted into electrical energy by the generator motor, and is stored in the battery. Here, in order to satisfy the above conservation law, all of the mass decreased in box ① must be an increase in the mass of the battery. In other words, all the lost mass becomes energy. It has a mass exactly equal to the mass whose energy has been reduced. This is regardless of the form of energy. The same is true for light energy, thermal energy, and chemical energy. This is because no matter how the energy is transmitted from objects A and B to the battery, the result will not change. Also, energy storage need not be limited to electricity, and various forms of energy storage do not affect the results. The same is true for devices that store light energy over long distances. Therefore, the photon also has a mass energy of hν/C^2. However, like the photon, the property of mass does not appear in the state of energy alone. The properties of mass appear when energy is confined in some kind of shell. A shell is usually a substance. →" True nature of gravity " Are you saying that? Even if Mr. A and Mr. B combine and release energy and the mass decreases or becomes close to zero, if they are separated again by the power generation motor, they will return to Mr. A and Mr. B with their original mass and individuality (reversibility). 2-5-3. Rejection of the myth (theory that the gravitational field is energy and momentum tensor) If the energy (mass) transferred to this battery had the ability to create a gravitational field, as Einstein's equations say, then the gravitational field would be transferred to the battery, which would be completely wrong . If so, when the mass of Mr. A and Mr. B decreases, the gravitational field will also become smaller, and it will become impossible to extract energy gradually. In that case, even if they are separated by a power generation motor, the attractive force between them is weak, so sufficient energy is not supplied, and Mr. A and Mr. B are simply separated. Then the energy does not flow back and cannot return to the original state. Mass remains reduced. In other words, it becomes irreversible. But this would be wrong according to the common sense of the laws of physics. Physical phenomena that are irreversible are time and entropy increase, and this case is a typical reversible change. If such an irreversible change were to occur, people would have to lose weight as they went up and down in the elevator. decrease the number of cells? Do cells get smaller? change personality? physically impossible. In such a strong gravitational field, Mr. A and Mr. B will die, so it is irreversible. In the thought experiment, A and B are very strong, so I'm done with it. Besides, even if it is an everyday weak gravitational field, the same thing as the above happens on a daily basis, even if the numerical value is small. 2-5-4. Whereabouts of gravitational field → attached to matter Then, correctly, the gravitational field must remain inside box ① without moving. From this, it can be concluded that the nature and ability to create a gravitational field are associated with Mr. A and Mr. B. The property and ability to create this gravitational field are called "true mass = gravitational elementary weight" here. In the past, gravitational elementary was treated as the same thing as mass energy, but here we will treat it as an independent ability. An object that has completely lost energy and mass can create a gravitational field, but since it has no mass, it will not receive force from other gravitational fields. In this way, the property and ability to create a constant gravitational field associated with an object is the gravitational elementary capacity. The unit of gravitational mass is kg for the time being. And it will be equal to the mass of matter that has not released any binding energy. This is the owner's hypothesis. 2-6. About the charge case Although the above proofs have mainly discussed gravitational coupling, similar conclusions can be drawn for charges. Here we consider the charge case in detail. In the case of gravity, the ability to create a gravitational field and the mass were conventionally regarded as the same. is the charge, and the mass is separated from the beginning, so the problem is clear. Substances (electrons, protons, etc.) contain electric charges, and strong electric forces act in electron-positron interactions, for example. It is easily assumed that the mass of the electron-positron will decrease if this electric binding energy is released, and that the mass will become zero if the binding energy increases further. This is the same conclusion as for gravity. However, in the case of electrons, etc., the experimental apparatus shown in Fig. 1 cannot be applied strictly. Attaching wires to electrons is out of scope for thought experiments. The binding energy cannot be extracted continuously, so it needs to be treated with quantum mechanics. However, the basic idea is the same as for gravity. The law of conservation of mass energy, the law of conservation of gravitational field, and the law of conservation of electric charge are applied. 2-6-1. Mass loss due to electron-positron coupling Empirically, electrons and positrons approach each other and release their binding energy by emitting photons (gamma rays ) . However, this emission is not performed continuously at any time, but two photons of 0.51 Mev equivalent to the mass conversion value of electrons and positrons are emitted and instantly combined. The electron-positron pair that released the binding energy becomes zero in volume xyz and mass in three-dimensional space, and seemingly ceases to exist. However, it continues to exist in 4D spacetime, and continues to have the ability to create a gravitational field. In addition, although the electric charge does not disappear, the electric force is canceled because the positive and negative equal charges are contained in the zero volume in the three-dimensional space, and the electric charge cannot be observed from the outside by normal methods. In the previous proof , we proved that mass and energy do not have the ability to create a gravitational field. I'm going to be We presume that this space contains many "electron: positron . These are not observed because they have zero volume and mass in three-dimensional space, but they have the ability to create a gravitational field, so the effect of this is likely to appear strongly, for example, in the rotation of the Milky Way. To explain the state of motion of most of the hundreds of billions of galaxies that have been confirmed so far , it is necessary to assume that such objects (dark matter) exist in greater quantities than ordinary matter. A huge black hole has also been observed in the early universe, but it is difficult for ordinary matter to accumulate on such a large scale in a short period of time. However, dark matter, which has no mass, travels at the speed of light, so it may have accumulated in the black hole in a short period of time and made its gravitational field enormous. Along with that, normal matter and light energy were absorbed and accumulated, and the mass became huge. This effect also caused galaxies to grow rapidly. 2-6-2. Reseparation of electron-positron combination (pair production) In vanishing masses (dark matter) such as electron-positron combinations, electric charges are conserved, but since the distance between both charges is almost zero, the electric charges cancel each other out, and the effect of the electric field to the outside is almost eliminated . but not strictly zero. Therefore, under special conditions, this charge pair may reveal its properties. For example, when it passes near the nucleus, it may change course or receive energy under the influence of the strong electric field. Under these conditions, when a high-energy photon or the like acts, it receives energy from it, breaks the bond, and generates an electron-positron pair. Here again, it is different from the idea of Mr. Dirac and others, but it can not be helped. I attach great importance to the laws of conservation of mass energy, electric charge and gravitational field. Existence should not arise from nothingness. And vice versa. 2-7. Experimental proof method A familiar example is iron, which is the most stable atom (having the highest binding energy), and is lighter than the sum of the protons, neutrons, and electrons that make it by the binding energy of the nucleus. Since the total mass of elementary particles is "true mass = gravitational elementary weight", iron is the substance with the largest difference between gravitational elementary weight and inertial mass. If the gravitational field created by 1 kg of such a substance and 1 kg of a substance with relatively small binding energy such as hydrogen is precisely measured, iron will have a slightly larger gravitational field. When an electron and a positron combine, a photon of 0.51 MeV x 2 and a mass of zero (electron-positron combination dark matter) are generated. If we can catch this, the validity of this theory will be proved. However, since it has zero mass, it moves at the speed of light, and since it has zero three-dimensional volume, it can pass through substances, making it difficult to detect. It's probably only detectable by the gravitational field it creates, but it's extremely small. This space contains a lot of vanishing mass dark matter, so if you create a high-energy state in the space, the bonds between the vanishing masses will break and a pair of positive matter and antimatter will be generated. If this proves it, the existence of the vanishing mass has already been proved. 3. Consideration: Conservation Law of Inertial Mass and Gravitational Field The theoretical development in this paper is based on the premise that the conservation law holds that the inertial mass and the gravitational field measured from the outside of Box 2 will not change no matter what reaction occurs in Box 2, which has no ingress and egress from the outside. I have to. If this is correct, the conclusion here should be correct. But if not, the conclusion here would be a big mistake. But if we assume that this conservation law is wrong, another big problem arises. For example, in the thought experiment device in Figure 1, suppose that the mass does not change even if the battery is charged. Consider the case where the conservation law does not hold. First, move A and B closer and farther apart in box ①. If the generator motor or anything else is 100% efficient, no external energy supply is required for this operation. Then, when looking at box ② from the outside, it is observed that the mass is changing. For example, if you put this on the ground, the weight will change repeatedly. If there is such a thing, for example, if it is placed on a spring, it will repeatedly move up and down. Then you can generate electricity with that movement. In other words, an infinite amount of energy can be extracted from nothing. This is a perpetual motion machine of the first kind and violates the first law of thermodynamics. At this stage, we can completely deny this as "impossible". In other words, the conservation law that the mass does not change when there is no going in and out of the box ② must be strictly established. Then we can conclude that the proof of the existence of dark matter, which we concluded in this paper, is also correct. In addition, it is concluded that the conservation law that the gravitational field does not change is also correct in the same process. For example, if the pair annihilation and pair generation of electrons and positrons causes their gravitational fields to vanish and generate, the result would be to allow the existence of a perpetual motion machine of the first kind, and the idea is strongly denied. . To explain this in detail, it is possible to assume an equilibrium state in which pair annihilation and pair generation occur repeatedly even if there is no input or output of energy in the box. If annihilation causes the gravitational field to disappear, the gravitational field of the box will change. Then, when a spring is placed on top of it and a substance is placed, the substance repeats up and down motion. Then, energy can be extracted from it, and it becomes a perpetual motion machine of the first kind. Therefore, such a thing cannot happen. In other words, even if they annihilate, we can conclude that their gravitational field does not change. The mass of an electron is very small, and the gravitational field it creates is at an unobservable level, but the fact that a first-class perpetual motion machine can theoretically be created no matter how small the output is, proves that assumption is wrong. become. At least in classical physics Strictly speaking, this operation changes the center of gravity position. Then, the energy will come out as gravitational waves, and it is also possible to take out the energy outside using the movement of the center of gravity inside the box. Therefore, from Box 2, the premise that "no energy goes in and out" cannot be fully observed, and the conservation law does not hold strictly. However, this does not affect the conclusions of this paper. Repeating the approach and separation of A and B means that the required energy is not strictly zero. As a thought experiment, this problem can be avoided by assuming that the box or apparatus is infinitely small.

A couple of weeks ago, I already wrote about me dealing with Galois theory and about how you can prove the fundamental theorem of algebra using its methods. Since then, I had a closer look at the geometrical implications of Galois’ work. The mathematical folklore is about the classical contructions with ruler and compass, what can and what cannot be done – in particular, every basic text on field theory talks about the impossibility of squaring the circle and doubling the cube and trisection of angles. All very nice and honestly: it’s actually surprising that you can disprove those geometric things with merely algebraic tools. Just to summarize briefly what’s the core of those folklore arguments: With a ruler and compass you can construct lines and circles (obviously) from certain given points, and by intersection, you get new points. When you consider equations for the circles and for the lines, you will see that you only have to deal with at most quadratic equations. Hence, every new point you construct with ruler and compass will come from a field extension of degree or . So, if you can construct something with ruler and compass, it must be contained in a field extension of that has degree for some . But then, for squaring the circle, you need to construct which is not algebraic; for doubling the cube, you need to construct , which has degree over ; and for trisection of an arbitrary angle you need numbers of degree as well. Contradiction. q.e.d. When the Greek mathematicians tried to solve their classical problems, they didn’t make it with only their classical tools (well… how could they, it’s impossible anyway). Instead, they invented many clever ideas to achieve a solution with slightly extended tools. One of these ideas is the famous quadratrix, which is a curve in the plane that crosses the -axis at distance from the origin (modernly speaking). With the usual tools, one can now construct and so, one has squared the circle. Similar things happen with the sprial of Archimedes. Something that I have encountered for the first time now, is the so-called neusis. I found it is sometimes also called a construction with compass and marked ruler. The classical Greek ruler is nothing but a stick, it only allows you to draw a straight line. Now that marked ruler has two marks at a certain distance of one another. At first, I wasn’t excited by this: if the distance of the marks is a constructible number, the compass already allows you to draw a straight line of this given length – now big deal about that. But the power of this tool is different: Suppose you have a point and two lines. With the marked ruler, you can draw a line through the given point that crosses the two given lines such that the segment between the two lines has the given length. You can’t do this with the classical unmarked ruler and compass alone, because you can’t know what angle the new line to be drawn needs to have! The marked ruler can be shifted until you have found the correct angle and draw the line then. Now, we have a new tool that comes quite cheap: Just mark the ruler in some fashion, for instance with the compass set to some unit length. But let’s see how powerful this can get. Let’s say the marks on our ruler have distance , it doesn’t matter if is constructible or not. We will trisect a given angle . To fix notation, is set at a point . Let’s draw a circle of radius around (see the picture below), such that one of the lines defining can be considered the diameter of the circle. The other line that defines will cut the circle at a point . Now, we set our marked ruler on the point and move it such that one of the two marks is on the circle, the other is on the diameter of the circle (the line through ). Thus, we have found a line segment of length between circle and line, passing the circle at point and the line at point ; and the prolongation of this line also passes through . The angle at between the new line and the diameter of the circle equals exactly . Well. Really? Let’s see. Of course the length r is present all over the place. It’s the distance of , of (since and are on the circle) and of (by marked-ruler-construction). So the triangle is isosceles and the angles at and at are equal. Similarly the triangle is isosceles with equal angles at and , let’s call this angle . The third angle of at equals . But this angle is also . So, and hence . Then consider the angles at : one of them is , the other one is . But this other one is also . So we have . Taking everything together, we find . q.e.d. This is a tricky construction, but it’s pretty much just moving around triangles. I can see how you can invent this trick. But those Greeks have found many more deep constructions using the marked ruler, just just look like magic to me. I can prove them, after I have thought about them long enough – but getting the idea to look at just that, this seems amazing. Besides, my proofs heavily rely on algebraic notation and on theorems that can’t possibly have been clear to the ancient Greeks. Maybe they just believed in the truth of those theorems and accepted them – they wanted to construct points after all, so they wanted to draw things; not invent any deep theory of algebra. But that doesn’t diminish their efforts and insights. I am very interested in the way they proved their results, sadly I couldn’t find any readable account of this. Let’s try to find cubic roots, for instance, with . We start with an isosceles triangle , where and . Prolonging the line beyond , we find the point with . Then, we draw a line through and , and we prolong the line beyond . Then, we can use our marked ruler through the point such that we’ll find a segment between the lines through and through with length . This segment cuts at the point and at the point , . Then, , amazingly. Once you have accepted this, you can find cube roots of any by scaling with a factor (and by constructing first). The tricky part is to prove the correctness of this construction in the first place. I don’t see how you could get this idea anyway… but it works just fine as you will see. We’ll need two new lines for our proof: A line at a right angle to through , meeting in ; and a parallel line to through , meeting the line in . Then the triangles and are similar, so we have and thus . The triangles and are similar as well (by the parallel lines and the respective angles at being equal). So, and hence . Now, we can use the Pythagorean theorem twice in a row: . This yields . By multiplying out and sorting all the terms, we find , and looking at this long enough, this is . Now, the first term gives , but this is negative and , being a line segment, needs to have positive length. So, the length of needs to be a positive solution to the second term, and so , as claimed (the other two solutions are not real). q.e.d. Isn’t that a pretty proof? Algebraically, by the way, the marked ruler allows you a little more than the classical ruler: It gives you points in field extensions of degree and . It still doesn’t allow you to construct any real number you want. The quadratrix on the other hand only gives you a field extension with the transcendental number , no cubic roots. Finally, I need to give credit to the book that showed me these things. The book by Cox on Galois theory, that I mentioned a couple of weeks ago already, has these things in an optional section and mostly as exercises. I have spent many an evening recently on those exercises – this was a lot of fun in geometry of all things (I have never liked geometry… at all.). The images that I have built in here, rely heavily on Cox’ notation and on the images to his exercises.

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THE PHYSICAL CONTENT OF h‑SPACE THEORY Classical mechanics is based on Newton’s laws and the principle of Galilean relativity. Let us consider each of Newton’s laws in terms of the proposed theory. Newton’s first law says. “Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed(from the original Latin of Newton’s Principia translated to English; http://en.wikipedia.org/wiki/Newton’s_laws_of_motion).”Given that atoms are made up of n=0-objects(II)”−”,”+”, the laws of classical mechanics describe the relative motions of the n=0‑objects(II)”−”,”+”. The velocity of n=0-objects(II)”−”,”+” is constant in the absence of their attraction or repulsion and the generation of n=1-, n=2- and n=3-objects. Consequently, the Newton’s first law express nothing but the constancy of the velocity of n=0‑objects(II)”−”,”+”. The second law of Newton states the following. “The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed (from the original Latin of Newton’s Principia translated to English; http://en.wikipedia.org/wiki/Newton’s_laws_of_motion).”In modern physics, a force is defined as the product of the acceleration and the inertial mass. In the proposed theory, the concept of inertial mass is defined as the amount of n=0-objects(II)”−”,”+”. Gravitational mass has the same definition. The direct proportionality of gravitational and inertial mass is due to the definition of mass as the amount of n=0-objects(II)”−”,”+”. Acceleration is a change in velocity per unit of time. The time unit, as discussed in the first pages of the theory, is a unit of velocity. Accordingly, the concept of force is a secondary concept, which is more convenient for determination of motion with variable speed, but it can be reduced to a derivative of the velocity. More simply stated, the concept of force is nothing more than another designation of a change in velocity. In the proposed theory, the central force acting on an object (as in the case of gravitational and electrostatic interactions) is defined in terms of the change of velocity due to the density gradient of n=0-objects(I) around n=0‑objects(II)”−”,”+” (electrons, positrons). The gradient is determined in inverse proportion to the square of the distance, which is due to the three-dimensionality of space. In other words, accelerated motion, i.e. movement with increasing velocity, is not a result of a force acting on a body but rather a consequence of its location in a region of space with a characteristic density gradient of n=0‑objects(I). Since distance is expressed in absolute units, i.e. in lengths of n=0‑objects(I), then velocity varies discretely. This differs from classical mechanics, in which there is no discreteness and velocity changes continuously. The concept of inertial forces is consistent with the suggested theory, since it describes nothing more than motion with variable velocity by changing the direction of motion. There is then no need for a source of inertial forces, as the concept of force is an auxiliary concept, in contrast to the fundamental concept of velocity. Newton’s third law states. “To every action there is always an equal and opposite reaction: or the forces of two bodies on each other are always equal and are directed in opposite directions (from the original Latin of Newton’s Principia translated to English; http://en.wikipedia.org/wiki/Newton’s_laws_of_motion).”In the proposed theory, this law is due to the symmetry of repulsion of n=0‑objects(II)”−”,”+”. In classical mechanics, momentum is defined as a product of velocity and inertial mass, and the momentum conservation law establishes the equality of a product of velocity and inertial mass for the closed system in two different states. Inertial mass determines the number of n=0‑objects(II)”−”,”+”. Thus, the law of conservation of momentum reflects the fact that the change in velocities of n=0-objects(II)”−”,”+” in a closed system is a result of a redistribution of the velocities of n=0‑objects(II)”−”,”+”. In the suggested theory, the momentum conservation law is valid for a system consisting of only n=0‑objects(II)”−”,”+”, i.e. electrons and positrons. In contrast to the modern physics, the interaction of the photon and electron causes no redistribution of momentum. The photon is absorbed by an electron, leading to an increase in electron velocity in the direction of the photon. If the photon is reflected from the positron, the positron will not change its velocity. Thus, the law of conservation of momentum cannot be applied to electromagnetic quanta. This statement is supported by the results of experiments with EmDrive (Harold White, Paul March, James Lawrence, Jerry Vera, Andre Sylvester, David Brady, and Paul Bailey 2016, Measurement of Impulsive Thrust from a Closed Radio-Frequency Cavity in Vacuum, Journal of propulsion and power, pages: 1-12, DOI: 10.2514/1.B36120). In contrast to momentum, energy is proportional to the square of velocity, and is not a vector, but a scalar. This concept of energy is different from that used in the proposed theory, in which energy is the opposite of length. Why is the classical concept of energy defined by the square of velocity? We consider this is due to the orthogonality of Euclidean space. In such space the square of velocity is a scalar form of velocity, which allows the algebraic operations of addition and subtraction in the three-dimensional space, i.e. along three orthogonal axes. Accordingly, describing the energy of n=0-object(II)”−”,”+”, i.e. square of the velocity of n=0‑object(II)”−”,”+”, enables one to express the conservation of n=0‑objects(II)”−”,”+” velocities in the form of a scalar sum of squares of velocities of n=0-objects(II)”−”,”+”. The proposed theory is entirely consistent with the laws of classical mechanics. It holds the principle of relativity of Galileo and Newton’s laws. In contrast to classical mechanics, the proposed theory provides an unambiguous definition of mass as the number of n=0-objects(II)”−”,”+” (electrons/positron). The concept of force is defined as an auxiliary, secondary notion. The motion of n=0-objects(II)”−”,”+” is primary, and change of their velocity defines the concept of force (for the given amount of moved n=0‑objects(II)”−”,”+”). The laws of conservation of momentum and energy are consequences of the primacy and constancy of motion of n=0-objects(II)”−”,”+”. In contrast to classical mechanics, in the proposed theory velocity changes discretely in potential fields: gravitational and electrostatic. I.e. in these fields there is a minimal change of velocity. SPECIAL THEORY OF RELATIVITY Special Theory of Relativity (STR)was built as an electrodynamics of moving bodies based on two postulates. One of them is the constancy of the speed of light in all inertial frames of reference, or, in other words, the independence of the speed of light from the motion of the light source. The second is the principle of relativity. It is assumed that all inertial frames of reference are equivalent and there is no special frame of reference for the laws of electrodynamics – Maxwell’s equations, or for the laws of mechanics. Why did these postulates and STR based on them appear in physics? At the time when STR was created, the existence of an ether, an all pervading, fundamental reference frame, was being actively discussed. The Michelson-Morley experiment gave negative results in the detection of an ether wind (light was considered as a wave in the ether, and the ether wind would change the speed of light). If an ether had been detected, then it would have represented a special frame of reference for the phenomena of electrodynamics, and the STR would not have been necessary. However, in the absence of an ether, all inertial frames of reference were considered to be equivalent and therefore the Maxwell’s equations had to maintain their form in different frames of reference, similar to classical mechanics. Experimentally, this was not found to be the case. There was asymmetry between frames of reference due to the fact that, for example, the speed of charge current and, consequently, the existence of a magnetic field depended on the choice of the frame. Also, the speed of light was supposed to be constant on the basis of experimental data, but the use of Galilean transformation required changes of it. Instead of Galilean transformation, Einstein proposed to use the Lorentz transformation and by this means eliminated the asymmetry between different inertial reference frames. In this way, he implemented the principle of relativity and the postulate of the constancy of the speed of light. The proposed theory is consistent with the postulate of STR regarding the constancy of the speed of light, but in contrast to STR the constancy of the speed of light has a cause – the spatial dimension. Since light quanta are n=1-objects, i.e. objects of one-dimensional space, they are always moving at the same velocity relative to n=0‑objects(II)”−”,”+” of zero-dimensional space, regardless of the speed of n=0-objects(II)”−”,”+”. If the reference system is changed, the constancy of the speed is provided by the inverse relation of the length of photons, as expressed by the Doppler effect (see section on “Optics”). Further, if in Maxwell electrodynamics and so also in the STR, the electromagnetic field and electromagnetic quanta have the same nature and are defined by Maxwell’s equations, then in the proposed theory the electromagnetic quanta and the electromagnetic field are different entities of nature. The field consists of objects of zero-dimensional space, n=0-objects(I), while photons are the objects of one-dimensional space, n=1-objects. (As noted above, there is a connection between them. Electromagnetic field change can lead to the generation of n=1-objects.) Therefore, the speed of light in the Michelson-Morley experiment would not change relative to an ether wind because light is not a wave in the ether. In the proposed theory, the ether exists as moving n=0‑objects(I) creating three-dimensional Euclidean space. Because of this, an attraction/repulsion of charges (electrons/positrons) is defined in the Maxwell’s equations by the maximum density of n=0-objects(I) ρ0(for an electrostatic field) or its change ρΔ (for a magnetic field), generated by the motion of electrons/positrons. In contrast to the STR, the proposed theory posits that different frames of reference are not equal. I.e. the second postulate, the relativity principle, is not valid for phenomena of electrodynamics. There is a unique absolute frame of reference – an ether, the frame associated with the space of n=0-objects(I) moving relative to each other. Since the electrostatic field of the charge is defined by the density of n=0-objects(I),ρ0, it cannot be changed (without gravity) when the frame of reference is changed. In the absence of gravity, magnetic induction also does not depend on the choice of the reference frame, since it determines the change in density of directed n=0-objects(I), ρΔ, relative to undirected n=0-objects(I) of density ρ0. As suggested above, in Maxwell’s equations the constant ccorresponds to a product of velocity v0and maximum density of n=0-objects(I) ρ0, v0ρ0, and it is not the speed of light. Therefore, it should not obey Galilean transformation for velocity. The maximal velocity v0ρ0decreases with time, since density ρ0was greater in the past than at present. To date, it is comparable with the speed of light and corresponds to the constant c. According to the proposed theory, it is also wrong to use STR to describe the motion of a body consisting of atoms. If, in the case of photons, STR is consistent with the proposed theory, it is because of the photons length, as the objects of one-dimensional space, is changing in Doppler effects and keeping constancy of the speed of light. The same changes in length are not applicable to the objects of zero-dimensional space, n=0-objects(II)”−”,”+” (electrons and positrons) composing atoms. In other words, there is no Lorentz contraction of lengths of physical objects in the suggested theory. Also, STR is not suitable for explaining the dynamics of bodies composed of atoms. The increase in mass of a body with increasing velocity is not possible. The inertial mass is determined by the number of electrons/positrons, and this number cannot depend on the velocity of electrons/positrons. Therefore, there is no infinitely large mass at the speed of light. The velocity of electrons/positrons are not limited by the speed of light. In the early universe, the velocity of electrons/positrons was greater than the speed of light because the density of n=0-objects(I) ρ0was higher than today, giving them a higher velocity of electrostatic interaction. Another difference between the proposed theory and STR is the conception of time. In the proposed theory, as in classical mechanics, time is just a matter of agreement. Time is an artificial concept, introduced for convenience to describe the motion of objects, and must be defined as the same in all inertial frames of reference. In special relativity, because of the heterogeneity of time in different frames of reference, there are temporal paradoxes, such as the twin paradox, due to time dilation in the moving frame of reference. In the proposed theory, this is not possible. GENERAL THEORY OF RELATIVITY The concept of space-time is used in General Theory of Relativity (GTR) as it is in the special theory of relativity. In general relativity, the gravitational potential is identified with the space-time metric. Space-time is curved by a body having mass, and this causes a gravitational attraction. In the proposed theory, all phenomena take place in Euclidean space. Since both general and special relativity use the concept of space-time rather than Euclidean space, we consider these theories to be incorrect models to describe phenomena. Phenomena, which in modern physics are explained only by general relativity, have their own interpretations in the proposed theory. For example, the gravitational redshift in general relativity is explained by gravitational time dilation. In the proposed theory, it is explained by a decrease in the density of n=0-objects(I) ρ0around the gravitating body (see “Gravitational attraction”). The existence of black holes can also be explained by changes in density of n=0-objects(I) ρ0. Due to the high density of electrons and positrons, a density gradient of n=0-objects(I) will be formed where the speed of gravitational attraction is greater than the speed of light, causing photons to be unable to overcome this attraction. The same reason will also cause the gravitational delay of electromagnetic quanta – the effect of Shapiro. Another phenomenon predicted by general relativity is the gravitational deflection of light. In the proposed theory, this results from the same attraction as for the Shapiro effect. Because of the high speed of light, gravitational attraction of photons is not as noticeable as for slow-moving electrons and positrons (n=0-objects(II)”−”,”+”). The precession of the orbit of Mercury in the suggested theory has no obvious explanation. However, this value can be the result of the influence of other planets in the solar system, since in the proposed theory gravitational attraction has a maximal boundary. In the proposed theory, in contrast to the STR, the speed of a body is not limited by the speed of light. The velocity of electromagnetic interactions is limited by a maximum density of the vacuum particles, n=0‑objects(I),ρ0. The velocity determined by this density ρ0, v0ρ0, is comparable to the speed of light, the velocity of object of one-dimensional space. The objects of two- and three-dimensional space move at velocities greater than this, 10 and 20 orders of magnitude above the speed of light, respectively. What is considered in GRT as the curvature of spacetime, caused by a gravitating body, is the curvilinear motion in three-dimensional Euclidean space. For example, the deviation of light by the gravitational field is the result of the attraction of light quanta, as well as any physical body moving near another gravitating body. In the proposed theory, phenomena explained by relativistic and gravitational time dilation are interpreted differently. For example, atomic clocks were employed in the Hafele–Keating experiment to accurately measure time, but these devices are based on the emission/absorption of electromagnetic quanta, and such processes depend on the elementary charge. Accordingly, a change of elementary charge caused by a change in gravitational potential will affect electromagnetic emission/absorption in the atomic clock and so “change” the time. Because of this, the atomic clocks of GPS satellites must be calibrated. Similarly, the change of the elementary charge is responsible for the observed effect in an experiment with masers – in the Gravity Probe A experiment. Quantum mechanics was created to explain the atom, since a classical planetary model was not satisfactory and was valid only as the Bohr model. Without the Bohr postulates, the atom would have to die as a result of energy loss by electrons (in the form of electromagnetic radiation) and their collapse into the nucleus. This paradox was formally resolved by quantum mechanics, where electrons were not allowed to have trajectories. Quantum mechanics has parameters related only to the initial and final stationary states of the electrons in atom, but not to any trajectories. Instead of the coordinates and velocities of the electron, probability values were used to describe these stationary states. The proposed theory of the atom returns to the classical description of electron motion as having a trajectory. Electrons can move closer to the nucleus from more remote positions. At the closer distance, a photon can be generated if it complies with the integer value of the energy of the emitted photon. In this case, emission of a photon reduces the velocity of the electron. Ultimately, the electron can occupy the minimum distance at which its velocity becomes equal to zero relative to the nucleus. This distance is ≈ 10−10 m. If there is no such compliance, then the electron will not radiate, and will continue to move to the other side of the nucleus, and away from it until its attraction to the nucleus leads to a complete stop and subsequent reversal back to the nucleus at a speed corresponding to a given distance from the nucleus. In contrast to the planetary model, the proposed theory predicts that electrons do not have strictly defined orbits, but rather changing trajectories. This distinguishes also the proposed theory from quantum mechanics, where there is no concept of electron trajectories in the atom. One can say that the electron oscillates around the nucleus, with a maximum distance from the nucleus of ≈10−5 m. At the distance ≈10−10 m, the electron is at rest relative to the nucleus. The absorption of a photon by an electron will increase electron velocity and distance from the nucleus. Another problem that was solved by quantum mechanics is the wave-particle duality. When particles pass through a thin metal film, diffraction rings are formed on a screen behind the film. A similar pattern is observed in the case of X-rays. Since X-radiation is believed to be a wave, it was suggested that particles can act in a similar manner, like waves. This phenomenon was termed the wave-particle duality. To explain this in quantum mechanics, it was decided to replace the notion of a trajectory with the concept of a superposition of states, more precisely, the superposition of probability of alternative states, i.e. probability for a particle to be at the same time in alternative states. In this context, the particle in each experiment can be detected with a certain probability, in one of these states. In the suggested theory, there is no wave-particle duality. Wave feature of particles in such phenomena as diffraction and interference are not due to the wave nature of the particles, but because of the generation of waves in the vacuum, which consists of n=0-objects(I) (see “Optics”). The cause of waves of the n=0‑objects(I) is the motion of photons or other particles, since they displace the n=0-objects(I). The displaced n=0-objects(I) move like a wave and the velocity of these waves is close to the speed of light (slightly higher). Waves of n=0‑objects(I) create waves of particles, moving relative to the n=0‑objects(I). This mechanism can explain the interference in the double-slit experiment, where the intensity of particles was set so low that only one particle could pass through the slits at any time. The same interference pattern is observed at both high and low particle fluxes. Quantum mechanics argues that this result is the inherent property of the particles, their nondeterministic, probabilistic behavior, according to the uncertainty principle. In the proposed theory, this interference pattern arises from the interference of waves of n=0-objects(I), displaced by moving particles. In this way, the interpretation of the two-slit experiment is returned to the deterministic view. The proposed theory also explains the discreteness of atomic spectra. Space, in the proposed theory, is composed a finite number of n=0-objects(I), so space is not infinitely divisible, i.e. matter is discrete. Since the number of n=0-objects(I), defining their density in the certain area of the space, is finite, then the difference of densities for different electron positions are also integers that determines the length of the generated n=1-object, as a multiple of an integer unit (see “Atoms and spectra”). In the proposed theory, explanation of the phenomenon of quantum tunneling does not require the uncertainty principle. As presented above (see “Superconductivity”), the tunneling of the electron has the same nature as superconductivity. It is due to the lack of interaction of the tunneling electron with the nucleus and electrons between 10−15 m and 10−10 m from the nucleus of atom. THE CASIMIR EFFECT The Casimir effect is the attraction of electrically neutral conductors and insulators. The distance, from which the effect becomes detectable, is a few micrometers. With decreasing distance the attractive force increases in inverse proportion to the distance in power of four. In modern physics, the effect is explained by quantum fluctuations of virtual particles of the electromagnetic field. In the proposed theory, the effect can be due to density fluctuations of n=0-objects(I). The fact that the length of a n=0‑object(I), ≈ 10−5 m, is comparable with the distance at which the Casimir effect begins to appear (several micrometers) fits with the proposed interpretation. The main difference between the proposed theory and the quantum theory is the deterministic character of physical phenomena and the rejection of their probabilistic nature. The uncertainty principle is not a principle of nature; at best it is a statistical description, at worst – it is a delusion. Quantization of physical quantities is a manifestation of the discreteness and finiteness of matter of our universe.

Physical unit . 146 high-quality, full-color pages; Printed single-sided on 80-lb coated paper ; Science units are not spiral-bound because so many pages need to be cut, removed, or copied; Complete PDF download. Digital copy of full unit for optional printing of consumable pages or copies for multiple children in the same household Our videos will help you understand concepts, solve your homework, and do great on your exams. Important Questions for CBSE Class 12 Chemistry – Kinetics PREVIOUS YEARS’ QUESTIONS 2015 Short Answer Type Questions [I] [2 Marks] Question 1: Write two differences between order of a reaction and molecuiarity of a reaction. Answer : Question 2: Define rate constant (k). Write the unit of rate constant for the following: (i) First […] Home AP Chemistry Unit #9: Applications of Thermodynamics ENE-5.A Explain whether a process is thermodynamically favored using the relationships between K, Chemistry primarily uses five of the base units: the mole for amount, the kilogram for mass, the meter for length, the second for time, and the kelvin for temperature. The degree Celsius (o C) is also commonly used for temperature. The numerical relationship between kelvins and degrees Celsius is as follows. A cumulative constant can always be expressed as the product of stepwise constants. There is no agreed notation for stepwise constants, though a symbol such as K L Formula in Hill system is K: Computing molar mass (molar weight) To calculate molar mass of a chemical compound enter its formula and click 'Compute'. In chemical formula you may use: Any chemical element. Capitalize the first letter in chemical symbol and use lower case for the remaining letters: Ca, Fe, Mg, Mn, S, O, H, C, N, Na, K, Cl, Al. Chemistry Stack Exchange is a question and answer site for scientists, academics, teachers, and students in the field of chemistry. Unit consistency in rate T is the temperature, T TPW = 273.16 K by definition of the kelvin; A r (Ar) is the relative atomic mass of argon and M u = 10 −3 kg⋅mol −1 . 2SO 2 (g) + O 2 (g) ↔ 2SO 3 (g) After the reactants and the product reach equilibrium and the initial temperature is restored, the flask is found to contain 0.30 mole of SO 3. Based on these results, the equilibrium Year outline of the new NGSS 3-course model "Chemistry in the Earth System" embedded with earth science standards. Links to lesson segment bundles being uploaded. K_\text c K c. K, start subscript, start text, c, end text, end subscript. and. Q. Q Q. Q. can be used to determine if a reaction is at equilibrium, to calculate concentrations at equilibrium, and to estimate whether a reaction favors products or reactants at equilibrium. Symbol. K. SO4. Charge. 1+ 2–. Unit 3 - Area of Study 2: How can the yield of a chemical product be optimised? a spreadsheet to manipulate data to illustrate the constancy of Kc at constant If you are absent, or missed part of the notes, or lost a worksheet or handout, this is the place to come. When 0.40 mole of SO 2 and 0.60 mole of O 2 are placed in an evacuated 1.00–liter flask, the reaction represented below occurs. 2SO 2 (g) + O 2 (g) ↔ 2SO 3 (g) After the reactants and the product reach equilibrium and the initial temperature is restored, the flask is found to contain 0.30 mole of SO 3. Based on these results, the equilibrium Having said all of that, it is still not uncommon for many chemistry applications at this level, to totally ignore the concept of activities altogether, and to still quote units for K. This can be problematic when one tries to justify units for ∆G° in the equation ∆G° = -RT lnK, for example, but it still goes on. The numerical value of K depends on the particular reaction, the temperature, and the units used to describe concentration. For liquid solutions, the concentrations are usually expressed as molarity. If you are absent, or missed part of the notes, or lost a worksheet or handout, this is the place to come. 2017-02-01 Entropy increases when matter becomes more dispersed. For example, the phase change from solid to liquid or from liquid to gas results in a dispersal of matter as the individual particles become freer to move and generally occupy a larger volume.Similarly, for a gas, the entropy increases when there is an increase in volume (at constant temperature), and the gas molecules are able to move UNIT V FUELS Rev.Ed. 2013-14 Engineering Chemistry Page 109 3. British Thermal Unit (B.Th.U.) : The amount of heat required to raise the temperature of one pound of water (454g) by one degree fahrenheit. Samarbetar stater i webbkryss Kelvin UNIT I. PHYSICAL CHEMISTRY:- a) Gas law and Kinetic Theory:- Ideal gas equation thermodynamic quantities of cell reactions (DG, DH, DS and K) - Over 27 Jun 2016 11 Rate = k[A][B]0 units of k = mol L−1 s−1 /(mol L−1 )1+0 = s−1 For the reaction aA + bB → cC + dD A.K.GUPTA, PGT CHEMISTRY, KVS going to C plus D, the rate law could be expressed, rate equals rate constant k times the concentration of A to the m and B to the n, where m and n are the order of Macromolecular Science Turns 100 by Christine K. Luscombe and Gregory T. Russell En. Chemistry and Environment Chemistry Applied to World Needs The unit is also cofounder and partener of the Institut Curie chemical Library Jennemann R, Dransart# D, Podsypanina K, Lombard B, Loew D, Lamaze C, 12 Apr 2010 The temperature 0 K is commonly referred to as "absolute zero." On the widely used Celsius temperature scale, water freezes at 0 °C and boils at RIKEN Center for Computational Science Medicinal Chemistry Applied AI Unit. Unit Leader: Watanabe C, Okiyama Y, Tanaka S, Fukuzawa K, Honma T.: 7 Mar 2019 We constructed a unit plan using AACT resources that is designed to teach the Describe how temperature affects the rate of a chemical reaction. between the equilibrium constant (K) and the reaction quotient (Q), as R = gas constant (dependent on the units of pressure, temperature and volume) R = 0.0821 L atm K-1 mol-1, if, (a) Pressure is in atmospheres (atm) The equilibrium constant, capital K, is a thermodynamic quantity. Having said all of that, it is still not uncommon for many chemistry applications at this level, to totally ignore the concept of activities altogether, and to still quote units for K. This can be problematic when one tries to justify units for ∆G° in the equation ∆G° = … The Seven SI Units: This figure displays the fundamental SI units and the combinations that lead to more complex units of measurement. It should be apparent that the move into modern times has greatly refined the conditions of measurement for each basic unit … An Atomic Assault: A "Nuclear Chemistry" Unit (PBL) for High School ChemistrySummary: A father is accused of assault with a radioactive substance. To go forward with a criminal case, investigators need to know what substance was used. Coach business card case in signature canvas billån selvom man står i rki lars johansson allstakan P=pressure of the gas (atm) K=Henry's Law Constant (atm/M) ***Watch the units some chemist use the equation S=KP (solubility= Henrys constant x pressure). 101.3 kPa = 1 atm. There are 1,000 pascals in 1 kilopascal. - Stress symptoms in females - Forensiker uppsala - Antagning uppsala ekonomi - Betald rast - Gavor avdragsgill The Curie temperature was increased from 430 K for x=0 to 443 K for x=0.4. 0.1, 0.2) crystallize in the tetragonal symmetry in the range 10–400 K and converts to cubic symmetry above 450 K. The unit cell Journal of Solid State Chemistry. Reaction. Another property of a In order to understand how the concentrations of the species in a chemical reaction change with time it is necessary We can write the rate expression as rate = -d[B]/dt and the rate law as rate = k[B]b . Then simply solve for Ea i 2 Jan 2021 Get the definition of the reaction rate constant in chemistry and learn about the factors or reaction rate coefficient and is indicated in an equation by the letter k. The units of the rate constant depend on the or The rate constant, k, for the reaction or enough information to determine it. In some cases, we need to know the initial concentration, [Ao]. Substitute this information 9 Nov 2017 Andselisk correctly identified the law of dilution and the name Ostwald is often connected with it. Kdissociation=α21−α⋅c.

A sextant is a highly accurate scientific instrument, used for measuring the angle between two objects. Its best-known use is in celestial navigation when you measure the altitude of a celestial body. Celestial navigation is not the only situation where you can use a sextant though. A sextant can be used for: - Measuring the altitude of celestial bodies - Finding a vertical sextant angle of a charted object - Measuring horizontal sextant angles of multiple charted objects - Using vertical sextant angles to find a clearing range - Using horizontal sextant angles to find a danger angle Sextants are simply a tool used for measuring angles. When used for navigation, you can apply those angles in any number of different ways to find your position and keep yourself safe. Finding a line of position by measuring the altitude of a celestial body The best-known use for a sextant is to find the altitude of a celestial body as part of a celestial fix. You use the sextant to measure the angles between the horizon and any celestial body. Any celestial body will work. You can use the sun; the moon; planets; or stars. As long as the body you choose in is the nautical almanac, you will be able to perform the calculations you require. A celestial fix is a complicated procedure, best explained with a video demonstration. I made the video below to cover the whole process, so you can watch that if you would like to know in detail how to find a line of position with a sextant. If you don’t want to watch, then we can summarise the method instead. The principle of a celestial line of position is that you measure the altitude of a celestial body using a sextant, and compare it to the altitude that you expect the body to be, according to your almanac. The difference between the observed altitude and the calculated altitude gives you a line of position that you can plot on the chart. Taking three observations gives you three lines of position, which can be used for a three-point fix. The precision of the sextant is needed because each degree of error would give you an error of 60 nautical miles in your position. Why is it useful to find a line of position from a celestial body? In navigation, you need to be able to find a line of position from a celestial body because it is the only non-electronic way of fixing your position when out of sight of land. The sextant is by no means as accurate as a GPS, but it is accurate enough to use as a backup if your GPS should fail. Using the sextant to measure the altitude of celestial bodies gives you the most accurate measurement that it is possible to get with a handheld instrument. Nothing else is able to give you the same accuracy. Finding a range by measuring the vertical sextant angle of a charted object Branching away from celestial navigation, you can use the sextant for terrestrial navigation as well. When you have a charted object with a known height, you can use the sextant and trigonometry to calculate your distance from the object, giving you a line of position. A good example is a lighthouse, where a nautical chart will tell you the height of the focal plane of the light. In the diagram above, you can use your sextant to measure the angle, θ. The height of the lighthouse can found from the chart. You’ll need to apply the height of the tide to the charted height depending on how your chart references heights. For example, if heights are referenced to HAT, you will need to use the following formula: Height = Charted Height + HAT – Current Height of Tide Once you have found the current height of the lighthouse above the water level, and have measured the angle, θ, with your sextant, you can use trigonometry to calculate the distance to the lighthouse. Range = Height of Lighthouse / Tan θ With the range, you can then plot a line of position using normal chartwork techniques. You have a range, and a charted object, so you know you are somewhere along an arc of constant range from the charted object. Using the sextant gives you precision when measuring a vertical angle that just wouldn’t be possible with other instruments. This is especially important when the height is small compared to the range. You can measure the angle of a lighthouse that it 10s of meters tall, even when you are a few miles away. The sensitivity and precision of the sextant are essential when measuring such small angles. Why are vertical sextant angles useful? Using a sextant to measure vertical sextant angles allows you to get two lines of position off a single charted object. You can calculate the range, and you can take a compass bearing. With two lines of position, you can get a reasonable idea of your location. It isn’t yet enough to confirm your position because for that you would need a third line of position. The only other way to get a range to use as a line of position is with radar. Radars are electronic equipment, that cost a lot of money to install. They are also subject to the same failures as all electronic equipment. Once you have the skills to use a sextant to get a vertical sextant angle, you give yourself even more tools in your arsenal for coastal navigation. Using horizontal sextant angles to find a line of position from two charted objects Turning your sextant on its side, you can use it to find horizontal sextant angles. These are exactly what they sound like. They are the horizontal angle between two charted features. You can use a horizontal sextant angle to plot a line of position. Horizontal sextant angles use circle theorems to plot lines of position. Specifically, the “angle at the centre theory”. In the image above, circle theory tells us that the angle, “a”, will be twice the size of angle, “b”. You have used your sextant to measure angle “b”, so you can use mathematics to calculate the size of angle “a”. Once you have found angle “a” you can use triangle theory to calculate the size of both angles “c”. You can then use angle “c” to plot onto your chart and find the centre of the circle. Using the centre, and one of the charted features, as the radius, you can plot the entire circle. The circle is your line of position. The horizontal sextant angle between two charted objects gives you a single line of position. You can use a third object to get three lines of position and a full three-point fix. You take the angles between: - Object 1 & object 2 - Object 2 & object 3 - Object 3 & object 1 An alternative to using three objects is to use the two objects as we did in the example. You can then use the bearings of both objects as well to give yourself a three-point fix. Why are horizontal sextant angles useful? Horizontal sextant angles are particularly useful because they give you the ability to fix your position independently of any other equipment. Using three charted objects, you can fix your position using your sextant on its own. This is especially useful because it will give you an accurate position fix even when your boat’s compass is not working. The sextant does not rely on any external inputs. It is purely a scientific instrument used to measure angles. Using the accuracy of a sextant allows you to plot extremely accurate lines of position using horizontal sextant angles. Finding a clearing range using a vertical sextant angle of a charted object While you can use a sextant to plot a line of position using vertical sextant angles, you can extend that use to find a clearing angle. The principle of clearing angles is that as you get closer to a charted object, the angle that you measure will get bigger. You can calculate the maximum angle that you want to keep yourself a safe distance away from a hazard. We already found a line of position using the angle, θ. As you are navigating closer to the coast, a clearing angle tells you the maximum that you can let θ get to while keeping yourself safe. We call that θ(max). If I am on my sailing yacht, I can periodically check the angle between the top of the lighthouse and the surface of the water. As long as it is less than my pre-determined θ(max), I know that I am in safe water. Why is it useful to work with vertical clearing angles? Using a sextant to find a vertical clearing angle is especially useful for coastal navigation. If you know that you need to keep 0.25 nautical miles from a lighthouse to keep yourself safe, you can just calculate the angle that you are looking for. It is then incredibly fast to measure the angle with your sextant and check it is less than your clearing angle. The major advantage is that you do not need to constantly go back and complete trigonometry to find your distance from the lighthouse. Again, the sextant is independent of all electronic navigational equipment making it an ideal backup for keeping yourself safe. Assuming you have planned clearing angles during your passage planning stage, you can very quickly cross-check the safety margin that your GPS is giving you. Using a horizontal sextant angle to find a danger angle with charted objects In the same way that we worked backwards to find a vertical clearing angle, you can do the same thing with horizontal sextant angles. The principles are the same as they were for calculating lines of position, but we work backwards to find a danger angle instead. In the image above, I have plotted a 0.5 nautical mile ring around a danger. I then plotted a circle which runs through the two charted objects and the very edge of the danger area. The angle, “a” is found by measuring between the lines joining the edge of the danger area and the charted objects. You then set your sextant the same as angle “a”. As you approach the danger, you can observe the two charted objects, measuring the angle between them. The further away you are, the smaller the angle will be. As you get close, the angle between the charted objects will get bigger. You know you are safe as long as the angle you measure stays smaller than angle “a”. Why is it useful to use horizontal danger angles? Horizontal danger angles are useful for coastal navigation in the same way vertical sextant angles are. They can keep you a safe distance away from dangerous water. Again, they are completely independent of all electronics. If you lose your GPS and your radar, horizontal danger angles keep you safe using only your sextant. With horizontal danger angles, the sextant isn’t the only tool that can accomplish it. You can simply set some dividers or anything else that can hold an angle to be the correct amount. Using a sextant will be one of the most accurate ways of doing it though. The thing with danger angles is that they do not need to be measured accurately. You only need to know when you have crossed the danger angle. The advantage of using a sextant, however, is that whenever you take a measurement less than the danger angle, you can use the accurate measurement to complete a horizontal sextant angle plot and get an accurate line of position. What else can you use a sextant for? Sextants are useful any time you need to measure angles. We have already discussed a few situations that make the most of the fine accuracy that a sextant can achieve. When accuracy isn’t too important, while you can use a sextant, other methods will be plenty accurate enough and allow you to get the job done quicker. For example, doubling the angle on the bow. This is a method for finding your range from a fixed object using triangle theorems. You can use a sextant to measure the angle from the bow to the charted object, but the precision given by a sextant would be considered overkill. When doubling the angle on the bow, you only need enough accuracy to know when the angle is doubled, to the nearest degree. The ship’s compass is usually accurate enough for this purpose. Likewise, you could use a sextant to measure the relative bearing of different vessels to determine if a risk of collision exists. Again, using a sextant is overkill in this sort of situation. While it will give you a very precise answer, if you are relying on the sort of level of precision that a sextant will give you, it is safe to assume the risk of collision does exist. Overall, the sextant is a precision instrument, used for measuring angles. When you need a high level of precision, it can be the ideal tool to use. When precision is not too important, other methods may be more efficient than the sextant.

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . . Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . . In this problem, we have created a pattern from smaller and smaller squares. If we carried on the pattern forever, what proportion of the image would be coloured blue? Anne completes a circuit around a circular track in 40 seconds. Brenda runs in the opposite direction and meets Anne every 15 seconds. How long does it take Brenda to run around the track? Can you see how this picture illustrates the formula for the sum of the first six cube numbers? Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring? A huge wheel is rolling past your window. What do you see? Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . . Can you make a tetrahedron whose faces all have the same perimeter? Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges. Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning? If you move the tiles around, can you make squares with different coloured edges? If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable. A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . . Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces? Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153? Show that among the interior angles of a convex polygon there cannot be more than three acute angles. Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw? Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares? Can you use the diagram to prove the AM-GM inequality? Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring? Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers? Can you maximise the area available to a grazing goat? Draw a pentagon with all the diagonals. This is called a pentagram. How many diagonals are there? How many diagonals are there in a hexagram, heptagram, ... Does any pattern occur when looking at. . . . Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100? How many winning lines can you make in a three-dimensional version of noughts and crosses? Can you discover whether this is a fair game? How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes? What size square corners should be cut from a square piece of paper to make a box with the largest possible volume? On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need? What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board? Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel? What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle? Take a line segment of length 1. Remove the middle third. Remove the middle thirds of what you have left. Repeat infinitely many times, and you have the Cantor Set. Can you picture it? Can you describe this route to infinity? Where will the arrows take you next? A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides? Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . . ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square. Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . . Have a go at this 3D extension to the Pebbles problem. ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR? Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite? Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all? The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . . Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . . In a right angled triangular field, three animals are tethered to posts at the midpoint of each side. Each rope is just long enough to allow the animal to reach two adjacent vertices. Only one animal. . . . A bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end?

Dmitrii Matveevich Sintsov Viatka (now Kirov), Russia BiographyDmitrii Matveevich Sintsov was born in Viatka (sometimes written as Vyatka) which was a large city in western Russia and the administrative centre of the Kirov Province. The change of name of this city to Kirov did not happen until 1934 when it was renamed after the Soviet official Sergey M Kirov. He attended the Third Kazan High School, graduating with the Gold Medal in 1886. Later in the year in which he graduated from the High School, he began his studies at Kazan University, graduating in 1890. This University, the result of one of the many reforms of the Emperor Alexander I, was founded in 1805, and was famed in mathematics by having Lobachevsky as its rector from 1827 to 1846. By the time Sintsov began his university studies he was already convinced that mathematics was the topic for him to concentrate on, and he became a member of the mathematics section of the Physics and Mathematics Faculty of the university. His lecturers in mathematics were A V Vasil'ev, F M Suvorov, V V Preobrazhenskii and P S Nazimov. He also took courses in astronomy with D I Dubyago. Sintsov's first research was on Bernoulli functions of fractional order and he carried this out while taking his fourth year undergraduate courses. His paper on the topic was published in the Notices of the Kazan Physics and Mathematics Society in 1890. This was a remarkable piece of work for a student at this stage in his undergraduate studies and it earned him a Gold Medal. Although Sintsov's interests moved away from the areas of his first scientific investigations, nevertheless he did undertake further research into Bernoulli functions and published further papers on this topic near the beginning of his career. Having made such an excellent start to his research, his "esteemed teacher" Aleksandr Vasil'evich Vasil'ev (1853-1929) recommended that he continue his studies at the University of Kazan with the aim of qualifying as a High School teacher. He spent three years, from the beginning of February 1891 to the beginning of February 1894, taking the necessary courses to obtain his teaching qualification. During this period he was being advised on research topics by Vasil'ev and, following his advice, he wrote his Master's Thesis The Theory of Connexes in Space in Connection with the Theory of First Order Partial Differential Equations. I A Naumov explains in :- The German mathematician A Clebsch was the first to investigate the theory of connexes in the period 1870-1872. He considered plane connexes i.e., plane geometrical objects, where the point-straight line combination was chosen as the basic element of the plane. Such connexes are termed ternary. Clebsch constructed the geometry of a ternary connex and applied it to the theory of ordinary differential equations.Sintsov was appointed to the staff of Kazan University and taught there from 1894 to 1899. After leaving Kazan, Sintsov taught at the Odessa Higher Mining School, then, in 1903, he was appointed to Kharkov University where he taught until his death in 1946. He took a leading role in the development of mathematics at Kharkov University and, for many years, he was President of the Kharkov Mathematical Society. This Society is one of the early mathematics societies, being founded in 1879. Following Vladimir Andreevich Steklov's presidency from 1902 to 1906, Sintsov took over as President, and held the position until his death forty years later :- Through Sintsov's initiative, the Kharkov Mathematical Society was deeply involved in the improvement of mathematical education in the schools of the Kharkov region. Sintsov also put considerable effort into maintaining the Kharkov Mathematical Society mathematical library which is still one of the most complete mathematical libraries in the Ukraine.Sintsov had an outstanding research record, and published 267 works during his long and productive scientific and teaching career. Of course through his many years of research his interests varied but the main areas on which he worked were the theory of conics and applications of this geometrical theory to the solution of differential equations and, perhaps most important of all, the theory of nonholonomic differential geometry. I A Naumov writes :- His classical work on the theory of connexes, of which he was one of the founders, and on nonholonomic differential geometry are well known far beyond the frontiers of our country.The book in which the articles (written by Ja P Blank who was a student of Sintsov) and appear, contains a selection of the Sintsov's major works on nonholonomic geometry. These were first published during the years 1927-1940 and include: A generalization of the Enneper-Beltrami formula to systems of integral curves of the Pfaffian equation Pdx + Qdy + Rdz = 0 (1927); Properties of a system of integral curves of Pfaff's equation, Extension of Gauss's theorem to the system of integral curves of the Pfaffian equation Pdx + Qdy + Rdz = 0 (1927); Gaussian curvature, and lines of curvature of the second kind (1928); The geometry of Mongian equations (1929); Curvature of the asymptotic lines (curves with principal tangents) for surfaces that are systems of integral curves of Pfaffian and Mongian equations and complexes (1929); On a property of the geodesic lines of the system of integral curves of Pfaff's equation (1936); Studies in the theory of Pfaffian manifolds (special manifolds of the first and second kind) (1940) and Studies in the theory of Pfaffian manifolds (1940). At Kharkov University, Sintsov created a school of geometry which became the leading school in this field in the Ukraine and has continued to flourish through the years still today being a leading centre. There he studied the geometry of Monge equations and he introduced the important ideas of asymptotic line curvature of the first and second kind. In 1903 he published two papers on the functional equation , now called the 'Sintsov equation,' which are discussed by Detlef Gronau in . He writes:- Sintsov gave in 1903 an elegant proof of its general real solution, which has the form , where q is an arbitrary function in one variable. ... [Sintsov] was the first who gave (in two papers ... in 1903) elementary simple proofs of its general real solutions. But before, it was Moritz Cantor who proposed these equations (there are two equations). In his journal 'Zeitschrift fur Mathematik und Physik,' ... he published [a note on them] in 1896. Cantor quotes these equations as examples of equations in three variables which can be solved by the method of differential calculus due to Niels Henrik Abel. ... The proof of Sintsov is much simpler and elegant.Sintsov also took an interest in the history of mathematics and one of the major projects which he undertook in this area was the detailed study of the work of previous mathematicians at Kharkov University. This work provides a fascinating account of the development of mathematics there from the founding of the university in 1805. The Ukrainian Academy of Sciences honoured Sintsov by electing him to membership on 22 February 1939. - I A Naumov, Dmitrii Matveevich Sintsov (his life and scientific and pedagogical work) (Kharkov University Press, 1955). - Ja P Blank, D M Sintsov (1867-1946), in Ja P Blank, D Z Gordevskii, A S Leibin and M A Nikolaenko (eds.), D M Sintsov, Papers on nonholonomic geometry (Kiev, 1972), 4-8. - Dmitrii Syntsov, Encyclopedia of Ukraine (Toronto-Buffalo-London, 1993). - D Gronau, A remark on Sincov's functional equation, Notices of the South African Mathematical Society 31 (1) (2000), 1-8. - List of the scientific works of D M Sintsov, in Ja P Blank, D Z Gordevskii, A S Leibin and M A Nikolaenko (eds.), D M Sintsov, Papers on nonholonomic geometry (Kiev, 1972), 286-293. - I A Naumov, Dmitrii Matveevich Sintsov on the 100th anniversary of his birth (Ukrainian), Ukrainskii Matematicheskii Zhurnal 20 (2) (1968), 232-237. - I A Naumov, Dmitrii Matveevich Sintsov on the 100th anniversary of his birth, Ukrainian Mathematical Journal 20 (2) (1968), 208-212. - I V Ostrovskii, Kharkov Mathematical Society, European Mathematical Society Newsletter 34 (December, 1999), 26-27. Additional Resources (show) Written by J J O'Connor and E F Robertson Last Update April 2009 Last Update April 2009

Journal of Manufacturing Processes 16 (2014) 551–562 Contents lists available at ScienceDirect Journal of Manufacturing Processes journal homepage: www.elsevier.com/locate/manpro Technical Paper Bitter coil design methodology for electromagnetic pulse metal processing techniques Oleg Zaitov a,∗ , Vladimir A. Kolchuzhin b a b Belgian Welding Institute, Technologiepark 935, B-9025 Zwijnaarde, Belgium Chemnitz University of Technology, Department of Microsystems and Precision Engineering, Reichenhainerstrasse 70, D-09126 Chemnitz, Germany a r t i c l e i n f o Article history: Received 2 March 2014 Received in revised form 2 June 2014 Accepted 15 July 2014 Available online 22 August 2014 Keywords: Bitter coil Magnetic pulse welding Design methodology a b s t r a c t Electromagnetic pulse metal processing techniques (EPMPT) such as welding, forming and cutting have proven to be an effective solution to specific manufacturing problems. A high pulse magnetic field coil is a critical part of these technologies and its design is a challenging task. This paper describes a Bitter coil design using a newly developed methodology for a simplified analytical calculation of the coil and complementary finite element models (FE) of different complexity. Based on the methodology a Belgian Welding Institute (BWI) Bitter coil has been designed and tested by means of short circuit experiments, impedance and B-field measurements. A good agreement between the calculated and the experimental design parameters was found. © 2014 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved. 1. Introduction In a certain range of thicknesses and materials combinations EPMPT are more competitive than of the same name conventional manufacturing methods. However a wide industrial use of the technologies is limited, partly due to a lack of compact engineering guidelines for a coil design. The main purpose of the present work is to develop such guidelines for a pulsed Bitter coil. Different types of coils categorized by Furth et al. can be used for EPMPT. On the basis of an analysis of manufacturing techniques, principal design solutions and performance characteristics of the above-named coils described by Lagutin and Ozhogin one can conclude that within tubular applications the Bitter coils have high reliability, manufacturability and maintainability. These are the key characteristics for an industrial implementation of the coils and factors which defined our choice to develop the calculation methodology for them. The coil is an assembly of the alternating conducting and insulating discs, each with a radial slit as shown in Fig. 1. The contact between the disks is realized due to their overlap. The Bitter coils can be used with fieldshapers (FS). Unfortunately a joint analytical treatment of the coil and a FS is hugely limited. However the FS can be partially taken into account, but in ∗ Corresponding author. Tel.: +33 630775462. E-mail addresses: [email protected] (O. Zaitov), [email protected] (V.A. Kolchuzhin). order to be brief in this article we are focused on the calculation methodology for the direct acting Bitter coil. The coil design is a complex task and mainly includes the determination of appropriate coil materials, sizes, the electromagnetic parameters such as an inductance, a resistance and the B-field as well as thermal and stress loadings. Most publications dedicated to the high magnetic coil design deal with pulsed coils having constant current density distribution which is according to Kratz and Wyder approximately realized in multi-layer multi-turn coils. Some of the relevant publications within the constant current density coils design are represented below. Wood et al. proposed an approach to a material selection for such a coil. Knoepfel suggested a methodology to calculate main electromagnetic parameters of the coil: the inductance, the resistance and the central field. Similar methodology to find the main design parameters of the coil and its strength was proposed by Dransfeld et al. . A relatively precise and complete design of the coil can be fulfilled in software developed by Vanacken et al. . The pulsed Bitter coils have the current density distribution which is approximately in inverse proportion to the inner radius and therefore the above-mentioned publications become irrelevant in the present case. Nevertheless methods of finding single design parameters of the pulsed Bitter coils are found in specialized literature. For example, the inductance of the Bitter coil can be calculated using a method proposed by Grover . Knoepfel suggested a formula to find the central field of the coil. Moreover, the most comprehensive physical and mathematical interpretations of the general design principles and different calculation techniques of http://dx.doi.org/10.1016/j.jmapro.2014.07.008 1526-6125/© 2014 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved. 552 O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562 known. Ideally the field distribution law in the gap coil-WP must be specified based on demands of an application. A step-by-step explanation of the scheme is represented below. Initial data for calculation: 1. Demanded parameters of the field: an amplitude magnetic field in the centre of the gap coil-WP Bmax and the rise time of the field are given. 2. A WP geometry characterized by an outer radius r0 , a wall thickness r and a work area length l as well as WP material properties represented by the conductivity , the heat conductivity , the specific heat capacity c, the yield strength y and the mass density m are known. 3. Pulse generator data such as the storable energy W, the maximum current amplitude I0 , the short circuit frequency f0 , the inductance Li and the resistance Ri are convenient to know for a simulation of a current pulse but this information is not obligatory and can be specified later during the design process. Fig. 1. A principal construction of the Bitter coil: 1 – Bitter plate, 2 – connecting lead, 3 – contact, 4 – current path, 5 – flange, 6 – insulator. the design parameters of the pulsed Bitter coils are given by Kratz and Wyder . Despite a sufficient, mainly academically orientated theoretical knowledge on the pulsed Bitter coils design, a simplified but complete industrial design methodology does not exist. The fact has prompted us to rework a thorough academic approach into a compact, industry-friendly methodology of the analytical calculation of the pulsed Bitter coil. This has been done by analysing an applicability of theoretical models describing electromagnetic, strength and thermal parameters of the coil and adjusting them to the present coil embodiment. A principal novelty of the methodology is that every design parameter is modified by asymmetry factors reflecting real geometry of the coil. An implementation of the asymmetry factors and a frequency-dependent resistance has improved precision of the methodology. Moreover several supporting FE models have been developed aiming to partly verify the analytical approach and to get a deeper insight into the design parameters. As it will be shown further the methodology of the analytical calculation is an effective tool for defining the main design parameters. Furthermore each step of the methodology can be fulfilled on a paper. Finally short circuit experiments, impedance and B-field measurements have been used for a verification purpose. A list of the symbols used in this article and the corresponding meanings is represented in Table 1. 2. Methodology of the analytical calculation The analytical approach can only be applied to the coil having cylindrical symmetry, which means that there is no change in geometry when rotating about one axis, and when magnetoresistance phenomenon, eddy currents, plastic deformations and thermal stresses are neglected. Additionally the field at each instant of time is calculated as the static field of the coil with a certain current density. With the limitations stated above the methodology can be schematically represented in Fig. 2. The present scheme assumes an approach to the coil design provided that the demanded magnetic field in the gap coil-WP, its rise time, WP geometry and parameters of the pulse generator are Material assignment. An insulating material represented by allowable working temperatures, dielectric and ultimate compression strengths, and a coil material described by the conductivity , the heat conductivity , the specific heat capacity c, the ultimate tensile strength UTS and allowable working temperatures initial Ti and final Tf must be defined. Maximum field in the gap coil-WP. It is known that the maximum achievable field in the gap coil-WP (FS-WP) must be at least 40 Tesla and the rise time must not exceed 25 s for the most of welding applications. Using an efficiency coefficient introduced by Wilson and Srivastava one can connect the fields in the gaps coil-FS and FS-WP. Coil geometry value assignment. An inner radius r1 is defined by the WP outer radius r0 and an insulation gap g which is typically 0.75–1.5 mm, a nominal length of the coil l0 is determined by the work area length of the WP l, while an outer radius r2 , thicknesses of a turn and the insulation between the turns h, as well as the asymmetry parameters ϕ, , of the turns can be defined using the parameters of the existing prototypes or arbitrarily. Finally a nominal number of turns N can be estimated. Auxiliary calculations. These are calculations of two form-factors of the coil ˛ and ˇ reflecting relations between the sizes of the coil, an “effective” number of turns allowing to transform the asymmetrical real to the ideally symmetrical coil, the skin depth characterizing an attenuation of electromagnetic waves in a conductor, the demanded current in the coil, a so-called filling factor describing a structure of conducting and insulating regions in the total cross-section and the material integral connecting physical properties of the coil material with an amplitude field and a pulse length. Design limitations. The maximum achievable field in the coil is mainly limited by two factors. The first is the mechanical strength of the coil depending on the ultimate tensile strength UTS of the coil material, its geometry and a distribution of the current density in it. The second factor is the thermal one and is determined by the thermal physical properties of the coil material, its geometry, the distribution of the current density in the coil, the allowable temperature range and the demanded pulse length. Both factors have to be considered in conjunction and the strongest factor defining the maximum achievable field must be selected. Finally the demanded field in the gap coil-WP Bmax and the maximum achievable field B0 are compared and a decision is made according to the scheme. Inductance of the coil. Geometrical parameters of the coil such as the inner r1 and the outer r2 radiuses, the length lcoil , the effective number of turns and a self-inductance factor (˛,ˇ) depending on the coil geometry and the current density distribution in the coil determine the inductance of the coil Lcoil . O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562 553 Table 1 List of symbols. Symbol Unit Description Symbol Unit Description Bmax B0 r0 r l c y m W I0 f0 Li Ri UTS Ti Tf r1 r2 ı j jth L R U H E T T s mm mm mm MS/m W/m·K J/kg·K MPa kg/m3 J A kHz nH mOhm MPa K K mm mm mm mm A/m2 A/m2 nH mOhm V A/m V/m Maximum demanded field in the gap coil-WP Maximum achievable field in the gap coil-WP Rise time of the field WP outer radius WP wall thickness WP work area length Electrical conductivity Heat conductivity Heat capacitance Yield strength Density Discharge energy Maximum current amplitude of a generator Short circuit frequency of the generator Inner inductance of the generator Inner resistance of the generator Ultimate tensile strength Allowable initial temperature Allowable final temperature Inner radius of the coil Outer radius of the coil Thickness of a turn Skin depth Stress-determined current density Thermally-determined current density Equivalent inductance of the coil and the WP Equivalent resistance of the coil and the WP Discharge voltage H-field Electric field strength h D N i b ϕ mm C/m2 Insulation thickness Electric displacement Nominal number of turns Nominal number of intermediate turns Nominal number of end turns Cut angle Contact angle Connecting angle Filling factor Form-factors Resistance of the coil. There are two approaches to the resistance calculation in the methodology. First approach is made based on the equation of Ohmic power in the coil and operates with the resistivity of the coil material , its geometry and the filling factor describing a structure of conducting and insulating regions in the total cross-section. This method does not take into account an increase of the resistance with frequency caused by skin and proximity effects. The second approach enables calculating the frequency dependent resistance and was adopted from induction heating technique. Simulation of an equivalent circuit. Having defined the resistance and the inductance of the coil and knowing the parameters of the generator, a current pulse can be easily simulated using a differential equation of damped current oscillations at the given initial conditions. The rise time T/4 and the amplitude of the obtained current pulse Im must be compared with the demanded rise time of the magnetic field specified in the performance requirements and the maximum current in the coil Imax found earlier. The calculated values must approximately fit the demanded values. Otherwise the previous calculation steps are repeated using a new geometry and a material until the aforementioned correspondence is reached. 2.1. Auxiliary calculations After the value assignment of the coil geometry, the inner r1 and the outer r2 radiuses, the thickness of the turn , the insulation thickness h, the nominal number of turns N and the asymmetry parameters ϕ, , are approximately defined. Practically the asymmetry parameters represent cuts and contact surfaces in the real coil (Fig. 3). Therefore the real coil to be manufactured with N nominal turns is obtained by adding the asymmetry parameters to the ideally symmetrical coil with turns. deg deg deg ˛ ˇ Effective number of turns Self-inductance factor Resistivity Calculated amplitude current through the coil Maximum demanded current through the coil Relative magnetic permeability Permeability of vacuum Angular frequency Stress-determined field Thermally-determined field Pulse shape factor Capacitance Damped angular frequency Inductance of the coil Resistance of the coil to the constant current Active resistance to alternating current Gap between the coil and the WP Material integral Permittivity of space (˛, ˇ) Ohm·m A A Im Imax H/m rad/s T T 0 ω B Bth C ωd Lcoil Rcoil Rac g FMat (Ti ,Tf ) ε0 F rad/s H mOhm mOhm mm A2 s/m4 F/m 1. Similar to Izhar and Livshiz method, “effective” intermediate M and end X turns can be found from the following expressions: M= 360◦ − (ϕ + 360◦ X= 360◦ − (ϕ/2 + 360◦ ) (1) + ) (2) Then total number of efficient turns consists of “effective” intermediate and endplates is found from expression (3): =i·M+b·X (3) Therefore the active length of the coil is found from (4): lcoil = · + ( − 1) · h (4) Now a simplification of cylindrical symmetry of the coil can be applied: the real Bitter coil with N nominal turns but with a symmetry breakdown is reduced to the ideal symmetrical coil having turns. The obtained value must be rounded up to an integer number. 2. According to Kratz and Wyder the form-factors of the ideal coil are found from (5) and (6): ˛= r2 r1 (5) ˇ= lcoil 2 · r1 (6) 3. If the current density distribution in the coil is described by a function f(r,z) = r1 /r, which is typical for the Bitter coils, than the filling factor is defined from expression: r2 = r1 r2 r1 dr dr 0 lcoil 0 f (r, z)dz (7) f (r, z)dz The numerator describes the current density distribution in the conductor and the denominator describes the distribution 554 O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562 Fig. 2. Graphical interpretation of the analytical approach. in the whole cross-section, taking into account the insulation. After integration of (7) a convenient analytical form is obtained: = · lcoil 5. The skin depth is a well-known characteristic describing an attenuation of electromagnetic waves in a conductor is found from (10): (8) 4. The maximum current in the coil can be approximated as (9): Imax Bmax · lcoil = · 0 (9) ı= 2· · 0·ω (10) O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562 555 Fig. 3. Intermediate and end Bitter plates. 6. The physical properties of the conductor material and the initial and the final temperatures define the material integral FMat (Ti , Tf ): Tf FMat (Ti , Tf ) = Ti m · c(T ) .dT (T ) (11) Higher values of FMat (Ti , Tf ) allows to generate higher fields and longer pulses. jth = 2.2.1. Strength limitation Various models allow calculating the strength of different types of coils. Gersdorf et al. developed the spatial-averaged model which averages properties of a conductor material and an insulator material and operates with a new material having these averaged properties. Mechanical stresses in a turn are found from the equilibrium equation including a volume Lorentz force, a tangential force and a force which is caused by a pressure difference on its inner and outer surfaces. Liedl et al. proposed the layer model considering mechanical properties of the insulator material and the conductor materials separately. This model does not take into account an axial force caused by the radial component of the magnetic field. In the present paper the “free-standing wire” model and its mathematical description suggested by Kratz and Wyder are used. The model assumes that there are no mechanical interactions between the turns. This means that mechanical loads are not transferred from one turn to another and only circumferential stresses occur in the turns as a response to the radial Lorentz forces trying to expand the coil. Assuming an infinite length of the coil analytical expressions for a calculation of the stress-determined current density and the field can be written as (12) and (13): B = 1 · r2 · 1 ln(˛) ln(˛) √ UTS · for the most of cold deformed aluminium alloys in order to prevent recrystallization and a loss of strength. Insulating materials can have even lower temperature limit which must be taken into account. Then a maximum achievable thermally-determined current density and corresponding field are found from (14) and (15): 2.2. Design limitations j = Fig. 4. Self-inductance factor (˛,ˇ) for an ideal Bitter coil with current density proportional to 1/r as a function of the shape parameters ˛ and ˇ . UTS / 0 0 (12) (13) 2.2.2. Thermal limitation According to Lagutin and Ozhogin the thermal limitation can be represented as an upper bound of a temperature rise in a skin layer during a pulse. Knoepfel considered an extreme limiting condition when the upper bound corresponds to the melting temperature of the coil material. In the present methodology the temperature rise during the pulse is not calculated directly, instead this value is assigned based on the coil material properties as it was proposed by Kratz and Wyder . For example, according to Mathers the upper temperature must not exceed 200 ◦ C Bth = FMat (Ti , Tf ) tpulse · ς 0 · · r2 · jth (14) (15) Finally according to Kratz and Wyder the maximum achievable field for the coil is determined by the stronger of the two above-mentioned limiting conditions and has a mathematical interpretation in a form of (16): B0 = Min(Bth , B ) · ln(˛) (16) When Bth < B the maximum achievable field is fully determined by the thermal limiting factor, in the opposite case of B < Bth the capabilities of the coil are limited by the strength factor. In accordance with the scheme (Fig. 2) next step of the methodology will be available if the demanded field Bmax in the gap FS-coil is less than the maximum achievable field found above. 2.3. Calculation of the inductance and active resistance of the coil Several methods for the inductance calculation were found in literature. Each method uses a set of similar fixed parameters but distinguishes itself from another by a specific term. Izhar and Livshiz suggested a formula which takes into account the skin effect for a coil and therefore reflects frequency-dependent inductance. An approach developed by Kalantorov et al. operates by a parameter depending on a ratio of the height of the turn and the mean diameter of the coil and more suitable for the inductance calculation at relatively low frequencies. Finally a comparison of the calculated and the experimental inductances showed that a method including a self-inductance factor (˛,ˇ) (Fig. 4) as the specific term proposed by Kratz and Wyder resulted in the best precision (17) Lcoil = 2 · 0 · r1 · (˛, ˇ) 4· (17) Practically the equivalent inductance of the coil and the WP is of particular interest and on the basis of a formula for a oneturn coil and a coaxial cylinder equivalent inductance proposed by 556 O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562 2.4. Simulation of an equivalent circuit As is well known damped current oscillations caused by a discharge of a capacitor bank in a circuit with an inductance and an active resistance is described by Eq. (24): d2 I(t) dI(t) + ω2 I(t) = 0 + 2ˇ dt dt 2 (24) On the basis of initial conditions (25) particular solution of the equation (24) is found from (26) q(t = 0) = CU and I(t = 0) = 0 (25) U −ˇt e sin(ωd t) Lωd (26) I(t) = − In accordance with the scheme the calculation is completed when the amplitude current in the coil Im and its rise time T/4 satisfy the corresponding demanded values Imax and . 2.5. Calculation of central magnetic field using Fabry formula Fig. 5. Contact resistance versus compressing forces: 1 – Cu-Cu contact, oxidized √ √ clean surface Ra 6.3; 3 – Contact between Cusurface Ra 6.3; 2 – Cu-Cu contact, √ Cr-Zn plates, oiled surfaces Ra √3.2; 4 – Cu-Cu, clean surface; 5 – Contact between Cu-Cr-Zn plates, clean surface Ra 6.3 . Shneerson , one can calculate the equivalent inductance of the Bitter coil as (18), when the conditions (19) are met: L˙ ≈ 2· · 1+ 2 ·· 4·g · 0 · r0 · g · a1 + ln r0 g ·r0 (18) 4·g In applications different than EPMPT one may be interested in the central field generated by the coil. A relation between the central magnetic field B01 , the Fabry factor G(˛,ˇ), the magnetic energy Wm and the inner radius of the coil r1 is reflected in the Fabry formula. According to Kratz and Wyder the Fabry formula for the Biter coils is given by expression (27). In turn the Fabry factor G(˛,ˇ) (28) represents the shape of the coil, the type of current density distribution given by the function f(r,z) = r1 /r and the self-inductance factor (˛,ˇ). (19) The first resistance calculation was made according to Kratz and Wyder based on the equation of Ohmic power (Joule heat) in the coil: P = I 2 Rcoil (20) B01 = G(˛, ˇ) = 0 · Wm · G(˛, ˇ) r1 2 · (˛, ˇ) (27) r 2 f (r, z) (r 2 + z 2 ) 3/2 drdz/ f (r, z)drdz (28) This is the resistance to a constant current neglecting the contact resistances between the plates: Rcoil = N 2 · · · r1 · ˇ · ln(˛) (21) There are contacts in the coil and their resistance can also be taken into account. As shown in Glebov et al. for copper contacts and different compressing forces between them this resistance is found from Fig. 5. Formula (21) does not reflect an increase of the resistance with an increase of the current frequency which results in an understated resistance while considering frequencies from the common magnetic pulse technology range (10–25 kHz). Nevertheless Slukhotsky and Ryskin proposed an analytical form to calculate the frequency dependent resistance: Rac = N · coil · 2 · · r1 ıcoil · (22) As it can be easily seen formula (22) defines the resistance of the conductor with a length 2··r1 and a cross-section ·ı. It should be noted here that the nominal number of turns N is used as a total current path length is counted and there is no need to keep symmetry as in the case with the inductance. Having the inductance and the resistance of the coil determined the parameters of any current course can be found. By analogy of the equivalent inductance the equivalent resistance of the coil and the WP can be calculated from the following: R˙ ≈ Rac + WP · 2 · · r0 ıWP · l (23) Using designations for the geometrical parameters of the coil and integrating right part of (28) along the cross-section of the coil a convenient form of the Fabry factor is obtained: ⎛ G(˛, ˇ) = lcoil r1 ⎜ 2 1 r ln ⎜ ⎝ 2 (˛, ˇ) lcoil · ln l r 1 coil r2 + + l2 coil r2 1 l2 coil r2 2 ⎞ +1 +1 ⎟ ⎟ ⎠ (29) The higher the values of G(˛,ˇ) and the magnetic energy Wm , and the smaller the inner radius of the coil r1 the stronger the field is obtained. Having the current course and the inductance of the coil defined its magnetic energy can be calculated: Wm (t) = Lcoil · I(t)2 2 (30) Finally, formula (27) is to be applied and the central field can be found. 3. FE modelling of the coil The classical theory of electromagnetism is fully represented by the Maxwell’s equations and complementary constitutive laws (Table 2).The present time-harmonic magnetic problem is described by Eqs. (32)–(34), (36) and (37). Time-harmonic magnetic problem means that H-field can be represented as the following: H(t) = H 0 ejωt (38) O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562 557 Table 2 Differential form of Maxwell’s equations and complementary constitutive laws. Maxwell’s equations divD = divB = 0 rotE = − ∂B ∂t rotH = j + ∂D ∂t Constitutive laws D = ε0 εE B= 0 H j = E (31) (32) (33) (34) (35) (36) (37) Omitting intermediate computations the diffusion equation for the H-field can be written in the form of: ∇ 2 H = jω H(t) (39) Solutions of Eq. (39) for a semi-infinite space, different boundary conditions and the initial condition of Hz (r,z) = 0 for 0 < r < ∞ can be found in Knoepfel . As shown in Meeker the problem is numerically solved in vector potential formulation and with the use of the Neuman boundary condition, i.e. when flux lines are perpendicular to the boundary of the problem domain. Input data of the models, their statuses, hardware configurations and solution times are given in Table 3. Fig. 7. Tangential component of B-field along the radius. one of the verification steps of the analytical model. Results of the 2D modelling are represented below. 3.1. 2D modelling of the coil in FEMM The analytical approach cannot describe a field distribution pattern in the coil. For example in welding this means that an estimation of a range of impact velocities is impossible. This task can be done by modelling of the coil in user-friendly FEMM software developed by Meeker and distributed under the Aladdin Free Public License. Moreover the FEMM model can be considered as a 3.1.1. Central B-field A visualization of the obtained magnetic field is represented in Fig. 6. The absolute values of the fields along the central radius and in the gap coil-WP are shown in Figs. 7 and 8 respectively. The simulation allows defining the magnetic field strength in every point within and outside of the working volume showing a significant advantage over the analytical approach. Table 3 Summary data of the FE-models. Model Software Input data f, [kHz] Im , [kA] , [MS/m] 2D FEMM 10 75 25 3D ANYS Emag Number of nodes Number of elements Hardware Solution time [h] 257 670 513 324 0.03 5 775 365 1 438 560 Intel® CoreTM i5-460 M CPU @ 2.53 GHz, 4.0 GB RAM, Microsoft® Windows® 7 Intel® CoreTM i7-3930 K CPU @ 3.20 GHz, 64.0 GB RAM, Microsoft® Windows® 7 Fig. 6. Contour plot of |B|-field in the coil. 6 558 O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562 Fig. 10. Current density distribution along the turns. Fig. 8. B-field in the gap coil-WP. 3.1.2. Current density distribution in a turn A visualization of the obtained current density distribution in the coil is plotted in Figs. 9–11. The current density distribution along the central parts of the turns is quite similar whereas across the turns it drastically changes at the corners. This is explained by the following effect: elementary currents in the corners are coupled with a smaller magnetic flux than in the central part of the turn and as a consequence the induced counter electromotive force in the corners is smaller than in the centre. Practically the simulation can be used for an optimization of the radiuses of the corners aiming a reduction of an excessive current density and therefore preventing a local overheating. The current density distribution in the lateral surfaces of the outermost turns is mainly influenced by the proximity effect which is a result of an induction of the eddy currents in the conductor by magnetic fields generated by the neighbouring conductors. The effect increases the resistance of the coil and intensifies with a decrease of the distance between the conductors and an increase of an amount of the turns. 3.1.3. Inductance and resistance of the coil The magnetic energy of the coil is computed by FEMM in accordance with formula (40): Wm = 1 2 BHdV (40) where the integral is taken over the problem domain. As the current flowing in the coil is known the inductance is found from (41): L= 2Wm I2 (41) Another method to derive the inductance is to use the “Circuit Properties” button. For the present case the “Circuit Properties” data is listed in Table 4. The inductance is defined by the Flux/Current ratio found for 3 “effective” turns while the resistance is given by the Voltage/Current ratio obtained for 5 “effective” turns as the total current path length is counted and there is no need to keep symmetry as in the case with the inductance. 3.2. Parametric study of the coil by 3D FEM analysis In the present study, we employed the commercial product ANSYS Emag (part of ANSYS Academic Associate license) as the most advanced tool of a complex 3D modelling of the coil in the Fig. 9. Contour plot of current density distribution in the coil. O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562 559 Table 5 List of equipment used in the experiments. Method Equipment Parameter determined Short circuit experiment Field probe RLC-bridge Rocoil FH-4015, Integrator IJ-1729, Oscilloscope TiePie, Handyscope HS3 EELAB measuring coil HAMEG HM 8118 Resistance R Inductance L B-field Resistance R = R(f) Inductance L = L(f) observed that the coil resistance decreases if the cross-section (the thickness and the contact angle) increases and if a current path length (the inner radius and the cut angle) shortens. In order to get a higher B-field, it is highly important to reduce Joule heat generation. 3.3. Sensitivity analysis Fig. 11. Current density distribution across the turns at a distance of 0.5 mm from the inner surface of the turns. linear harmonic regime . To accurately calculate the current density distribution, a FE model must have a mesh size at the conductor outer boundary smaller or roughly equal to the skin depth. This will lead to a very large number of elements, and therefore the model will become too demanding for a desktop PC. For instance, at the operating frequency of 10 kHz and considering pure copper with the resistivity of 2.7 × 10−8 Ohm·m, the skin depth is about 0.8 mm. To avoid a very large number of elements one need to use a boundary layered mesh which consist of thin, elongated elements at the boundaries of conductors. The boundary layered 3D mesh was generated using commercial pre-processing software ANSA . The mesh was created with the SOLID236 element using 3D edge-flux formulation. The computational air domain was truncated with a cylindrical volume with the outer radius 2r2 and the open boundary was modelled with a flux-parallel boundary condition. The contact resistance between the plates is neglected in the model. The electrical resistivity of aluminium and copper are 4.0 × 10−8 Ohm·m and 2.7 × 10−8 Ohm·m, respectively. The main objective of the 3D FE-simulation is to perform a parametric study taking into account six input geometrical parameters: inner radius r1 , outer radius r2 , turn thickness , connection thickness h, contact angle and cut angle ϕ. Goal functions are the coil resistance, the inductance, and the B-field. A mesh morphing is used partially for geometrical modifications and a reuse of the initial FE-model. The inductance of the coil was calculated from the magnetic energy using formula (30). The power Prms dissipated in a conducting coil body under the harmonic excitation can be calculated as: Prms 1 = 2 · j(x, y, z)2 dV A local approach is applied to perform a sensitivity analysis by taking a partial derivative of each output parameter Gj with respect to an input parameter pi . This method examines small perturbations and a one parameter at a time. The obtained sensitivities are depicted in Fig. 13. These sensitivities reflect the calculated parametric dependences. All sensitivities except for the inner radius have negative value. The outer radius and the thicknesses have minimum and maximum sensitivities correspondingly. 3.4. Conclusions to 2D and 3D FEM simulation The 2D numerical model (FEMM) has shown higher capabilities for description of the electro-magnetic design parameters than the analytical approach as it takes into account the eddy currents induced in the coil. Another important advantage of the numerical model over the analytical one is an ability to compute the field at any point of the space. Nevertheless the method doesn’t include any strength or temperature estimation of the coil and therefore cannot be used independently. Finally it can be concluded that the numerical 2D model may complement the analytical one provided that results of the calculation of each model are close. The 3D numerical model using ANSYS Emag is the most advanced tool of a complex 3D analysis of the coil as it can take into account the asymmetry parameters of the coil. Moreover the parametric and the sensitivity analyses are convenient way of the design optimization. On this step the analytical, 2D and 3D numerical inductances, resistances and central magnetic fields and frequencies are defined. In order to find out an accuracy of each of the models and to verify the analytical approach experiments are needed. (42) 4. Experimental verification of the methodology Fig. 12 shows variations of the resistance and the inductance with regard to a respective geometrical parameter. Green points correspond to the initial values of the input parameters. It is clearly For the experimental verification four complementary methods were used (Table 5). Current curves obtained during short circuit Table 4 Circuit properties. Parameter Value 3 “effective” turns Total current [A] Voltage drop [V] Flux linkage [Wb] Flux/current [H] Voltage/current [Ohm] Real power [W] Reactive power [VAr] Apparent power [VA] 5 “effective” turns 75,000 138.298+i8474.16 0.133535−i0.000749625 1.78047e−006−i9.995e−009 0.00184397+i0.112989 5.18616e+006 3.17781e+008 3.17823e+008 333.663+i19314.1 0.3045−i0.000877415 4.058e−006−i1.16989e−008 0.00444884+i0.257521 1.25124e+007 7.24278e+008 7.24386e+008 560 O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562 Fig. 12. Results of the parametric study of the Bitter coil. Fig. 13. Results of sensitivity analysis of the Bitter coil. experiments were the basis for a determination of the average resistance R and the inductance L of the coil. B-field was measured by the field probe developed by Electrical Energy Laboratory (EELAB), University of Gent, and finally the resistance and inductance of the coil were measured in a range of frequencies 20 Hz–150 kHz using an RLC-bridge. 5. Discussion The results of the analytical and the numerical calculations as well as the experimental parameters of the coil are summarized in Table 6. As it can be seen from the Table 6, the short circuit experiment and the RLC-bridge measurement resulted in different inductances and resistances. A possible explanation is that the RLC-bridge measurement describes a steady-state process with the constant resistance and the inductance while the corresponding short circuit values describe a transient process with the instantenious resistance and the inductance which are significantly influenced by characteristics of switches of a generator. Based on the above-mentioned facts one can conclude that the RLC-bridge measurement is more suitable for the verification of the FE models with the time-harmonic approximation of the coil behaviour. In reality the coil works in the transient regime which has not been modelled in the present work. Each generator is characterized by a unique transient behaviour and therefore it is challenging to build one model which can describe different transient processes. 5.1. Active resistance Active resistances obtained analytically and experimentally during the short circuit measurement are relatively alike which shows a good capability of the analytical approach to describe behaviour of the coil attached to the generator (Table 7). At the same time the resistance obtained during the RLC-bridge measurement O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562 561 Table 6 Summary of calculated and measured values. Method Calculation technique Analytical Numerical 2D FEM (FEMM) Numerical 3D FEM (ANSYS Emag) Verification technique Short circuit RLC-bridge Field probe Active resistance to alternating current Rac , [mOhm] Inductance L [nH] 6.84 4.44 3.48 1548 1780 1541 7.7 4.6 1503 2309 Table 7 Parameters of the generator. B-field [T] Central Outermost 1.36 1.46 1.44 2.3 2.17 1.5 1.9 Frequency of the field f [kHz] 10 10 10 Table 8 Discharge currents and relative errors. Parameter Value C [F] Umax , [kV] Ri [mOhm] Li [nH] f0 [kHz] 160 25 2.95 42.58 60.7 is close to both numerical 2D and 3D resistances which in turn verifies the numerical models. 5.2. Inductance Inductances obtained analytically, by both numerical methods and by the short circuit experiment are relatively close but smaller than the value from the RLC-bridge measurements. One of the possible explanations of the differences has been mentioned above. 5.3. Central magnetic field The analytical, both numerical and the experimental fields are close to each other, which is a positive verification fact. In order to find out if the differences between the calculated and the experimental parameters are appropriate (Table 6) behaviour of the coil connected to the generator is simulated by solving the differential equation of damped current oscillations (26) for the measured RLC-bridge and the analytically calculated parameters and a comparison of the obtained current curves with the short circuit experiment is made (Fig. 14). First quarters of periods of the experimental and each of the simulated damped current oscillations, practically defining a Method Rise time [s] Amplitude, current Im [A] Short circuit RLC-bridge Analytical Error relative to short circuit/RLC-bridge experiments [%] Error relative to B-field measurement [%] 29.5 28 25 6/10 75,077 78,824 91,018 23/15 9 technological effect on the WP, and errors of the analytical calculation relative to the short circuit current and the solution for the measured RLC-bridge values are represented in Table 8. The analytically calculated current shows the smallest errors relative to the solution for the measured RLC-bridge values, therefore this experimental technique is preferable in the present case. Based on this fact it can be concluded that the simplified methodology of the Bitter coil analytical calculation is verified on the basis of the RLC-bridge measurement with relative errors of 9% in the central field, 10% in the rise time and 15% in the amplitude current (Table 8). In the present work the equivalent inductance (18), resistance (22) and the frequency of the electromagnetically coupled coilWP system were not checked experimentally. In practice at the same discharge energy the coil-WP system will generate a higher frequency and a current than the coil used separately due to a smaller equivalent inductance and therefore obtaining the maximum allowable rise time less than 25 s only in the coil can overcome difficulties of the exact coil-WP inductance estimation and be satisfying. Moreover the maximum current amplitude Im increases with an increase of the charging energy of the capacitors. This fact can be used to compensate relatively small differences between the experimental and calculated current amplitudes. 6. Conclusions Fig. 14. Current curves obtained with short circuit, RLC-bridge and analytically defined inductances and resistances. The industry orientated methodology for the simplified analytical calculation of the pulsed Bitter coil has been developed. Coil asymmetry characteristics implementation allowed increasing a precision of the analytical models describing main design parameters of the coil. Additionally the 2D and the 3D FEM model have been developed aiming to partly verify the analytical approach as well as to get a deeper insight into the design parameters. Based on the methodology a Belgian Welding Institute (BWI) Bitter coil has been designed and tested by means of short circuit experiments, impedance and B-field measurements. Differences between the calculated and the experimental B-fields, rise times and amplitude currents were found to be 9%, 10% and 15% correspondingly, which allows drawing a conclusion of a positive verification of the methodology. Therefore the developed methodology can be 562 O. Zaitov, V.A. Kolchuzhin / Journal of Manufacturing Processes 16 (2014) 551–562 practically used for the Bitter coils design, including determination of geometrical, thermal and load-bearing boundaries of the coil and includes: 1. Iterative determination of the geometry and the material of the coil until they meet the requirements specification. 2. Calculation of the two main electromagnetic parameters of the coil: the inductance and the resistance which along with the parameters of the generator form a current pulse. 3. Determination of the maximum allowable magnetic field generated by the coil based on the thermal and the strength properties of the conductor material, the current density distribution in it and on the geometry of the coil. 4. Determination of the demanded current pulse satisfying the performance specification. Additional use of FEMM overwhelms some of the limitations of the analytical approach mainly due to its ability to calculate the field at any point of the space. FEMM can also play a role of a fast verification tool for such parameters as the B-field, the resistance and the inductance of the coil. Finally if time and resources are not limited ANSYS Emag can be used as an advanced analysing tool taking into account multiphysical interactions and the asymmetry parameters of the coil which makes this tool the most comprehensive in the design optimization and refining. Acknowledgments This work has been done within the ACODEPT (Advanced Coil Design for Electromagnetically Pulsed Technologies) funded by the European Commission within the CORNET programme. The CORNET promotion plan 61 EBR of the Research Community for European Research Association for Sheet Metal Working has been funded by the AIF within the programme for sponsorship by Industrial Joint Research (IGF) of the German Federal Ministry of Economic Affairs and Energy based on an enactment of the German Parliament. The authors would like to thank Mitko Bozalakov for help in conducting the impedance and B-field measurements and EELAB of University of Gent for providing us with the measuring equipment. References Dransfeld K, Haidu J, Herlach F, Landwehr G, Maret G, Miura N, et al. Strong and ultra-strong magnetic fields and their applications. Springer Verlag; 1985. Furth HP, Levine MA, Waniek RW. Production and use of high transient magnetic fields. Rev Sci Instrum 1957;28(11):949–58. Gersdorf R, Muller FA, Roeland LW. Design of high magnet coils for long pulses. Rev Sci Instrum 1965;36:1100–9. Glebov LV, Peskarev NA, Figenbaum DS [in Russian] Calculation and design of machines for contact welding. Energoizdat; 1981. Grover FW. Inductance calculations working formulas and tables. New York: Dover Publications; 1946. Izhar A, Livshiz Y. Non-destructive coils and field-shapers for high magnetic field industrial applications. Pulsar Electromagnetic Technologies; 2002. Kalantorov PL, Ceitlin LA [in Russian] Inductance calculations. Handbook. Leningrad: Energoatomizdat; 1986. Knoepfel H. Pulsed high magnetic fields. North Holland Publishing Company; 1970. Kratz R, Wider P. Principles of pulsed magnet design. Berlin/Heidelberg: Springer-Verlag; 2002. Lagutin AS, Ozhogin VI. Pulsed magnetic fields in physical experiments [in Russian]. Monographie. Moscow: Energoatomizdat; 1988, 190 pp. Liedl J, Gauster WF, Haslacher H, Grossinger R. Calculation of the mechanical stresses in high field magnet by means of layer model. IEEE Trans Magnet 1981;17(6):3256–8. Mathers G. The welding of aluminium and its alloys. Woodhead Publishing Limited; 2002, 236 pp. Meeker DC. Finite Element Method Magnetics, Version 4.0.2 (11Apr2012 Build); 2012 www.femm.info Shneerson GA. Proceedings of the Soviet Union Academy of Sciences [in Russian]. Energy Transport 1969;2:85. Slukhotsky AE, Ryskin SE. Inductors for induction heating [in Russian]. L. Energiya 1974:1974–2264. Vanacken J, Liang L, Rosseel K, Boon W, Herlach F. Pulsed magnet design software. Phys B 2001;294–295:674–8. Wilson MN, Srivastava KD. Design of efficient flux concentrators for pulsed high magnetic fields. Rev Sci Instrum 1965;36(8):1096. Wood JT, Embury JD, Ashby MF. An approach to materials processing and selection for high-field magnet design. Acta Mater 1997;45(3):1099–104. ANSYS® Academic Research. Release 14.0, Help System, Low-Frequency Electromagnetic Analysis Guide. ANSYS Inc.; 2011. http://www.beta-cae.gr/ansa.htm Oleg Zaitov received a bachelor’s degree and a master’s degree in welding engineering from Ufa State Aviation Technical University, Russia, in 2007 and 2009 correspondingly. During 2009–2010 he was working as a researcher in the field of linear friction welding at the Department of Welding Engineering at Ufa State Aviation Technical University. In 2011 he joined Belgian Welding Institute as a research engineer with the main focus on magnetic pulse welding (MPW) developments. His research interests are a development of mathematical models for preliminary weldability assesment in MPW, a coil design and its optimisation for MPW. He has been writing his PhD within these topics. Vladimir A. Kolchuzhin received a bachelor’s degree and a master’s degree in electrical engineering from Novosibirsk State Technical University, Russia, in 1997 and 1999, respectively. During 1999–2003 he was working as a Scientific Assistant at the Department of Semiconductor devices and Microelectronics at Novosibirsk State Technical University. In Nov. 2003 he joined the Department of Microsystems and Precision Engineering at Chemnitz University of Technology, Germany where in 2010 he received his Doctoral degree. He has been working in the field of the advanced modelling methods development for MEMS. His research interests are numerical methods for the nano- and microsystems design.

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes? In a three-dimensional version of noughts and crosses, how many winning lines can you make? How many different symmetrical shapes can you make by shading triangles or squares? How many different triangles can you make on a circular pegboard that has nine pegs? Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw? We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used? Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations? On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win? You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters? This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up? How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on! A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides? A game for 2 players. Given a board of dots in a grid pattern, players take turns drawing a line by connecting 2 adjacent dots. Your goal is to complete more squares than your opponent. Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . . Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square. Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too. A game for 2 players. Can be played online. One player has 1 red counter, the other has 4 blue. The red counter needs to reach the other side, and the blue needs to trap the red. An extension of noughts and crosses in which the grid is enlarged and the length of the winning line can to altered to 3, 4 or 5. Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers? An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length? A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard? Exchange the positions of the two sets of counters in the least possible number of moves Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back A game for two players on a large squared space. A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all? Can you fit the tangram pieces into the outline of the telescope and microscope? Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians? A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work? This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions. Can you fit the tangram pieces into the outline of these rabbits? Can you fit the tangram pieces into the outline of Wai Ping, Wah Ming and Chi Wing? Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need? What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one? ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square. Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . . ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR? Can you find a way of representing these arrangements of balls? Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper. Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours? Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex? Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th Can you work out what is wrong with the cogs on a UK 2 pound coin? Can you fit the tangram pieces into the outlines of these people? Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube? Can you fit the tangram pieces into the outlines of these clocks? Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts? Can you fit the tangram pieces into the outline of Little Fung at the table?

80 relations: Absorbance, Absorption (electromagnetic radiation), Absorption cross section, Air pollution, Atomic physics, Azimuth, Barn (unit), Beer–Lambert law, Centimetre, Classical mechanics, Collision, Common logarithm, Coulomb's law, Cross section (geometry), Dirac delta function, Elasticity (physics), Electromagnetism, Elementary particle, Femtometre, Flow velocity, Fog, Gas, Geometrical optics, Grammatical modifier, Gravity, Impact parameter, International System of Units, Lens (optics), Light, Logarithm, Luminescence, Luminosity (scattering theory), Magnetism, Mean free path, Meteorology, Metric prefix, Micro-, Micrometre, Milli-, Momentum, Momentum transfer, Natural logarithm, Nephelometer, Neutron, Neutron cross section, Nuclear cross section, Nuclear physics, Number density, Partial wave analysis, Particle detector, ..., Particle physics, Path length, Permittivity, Picometre, Quantum mechanics, Radar cross-section, Radius, Ray (optics), Reaction rate, Reduced mass, Resonance (particle physics), Rutherford scattering, S-matrix, Scattering, Scattering amplitude, Scattering theory, Sigma, Sine, Solid angle, Sphere, Spherical coordinate system, Square metre, Stationary state, Steradian, Transmittance, Transversality (mathematics), Visibility, Wave function, Wavelength, X-ray. Expand index (30 more) » « Shrink index In chemistry, absorbance or decadic absorbance is the common logarithm of the ratio of incident to transmitted radiant power through a material, and spectral absorbance or spectral decadic absorbance is the common logarithm of the ratio of incident to transmitted spectral radiant power through a material. In physics, absorption of electromagnetic radiation is the way in which the energy of a photon is taken up by matter, typically the electrons of an atom. Absorption cross section is a measure for the probability of an absorption process. Air pollution occurs when harmful or excessive quantities of substances including gases, particulates, and biological molecules are introduced into Earth's atmosphere. Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. An azimuth (from the pl. form of the Arabic noun "السَّمْت" as-samt, meaning "the direction") is an angular measurement in a spherical coordinate system. A barn (symbol: b) is a unit of area equal to 10−28 m2 (100 fm2). The Beer–Lambert law, also known as Beer's law, the Lambert–Beer law, or the Beer–Lambert–Bouguer law relates the attenuation of light to the properties of the material through which the light is travelling. A centimetre (international spelling as used by the International Bureau of Weights and Measures; symbol cm) or centimeter (American spelling) is a unit of length in the metric system, equal to one hundredth of a metre, centi being the SI prefix for a factor of. Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies. A collision is an event in which two or more bodies exert forces on each other for a relatively short time. In mathematics, the common logarithm is the logarithm with base 10. Coulomb's law, or Coulomb's inverse-square law, is a law of physics for quantifying the amount of force with which stationary electrically charged particles repel or attract each other. In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. In mathematics, the Dirac delta function (function) is a generalized function or distribution introduced by the physicist Paul Dirac. In physics, elasticity (from Greek ἐλαστός "ductible") is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles. In particle physics, an elementary particle or fundamental particle is a particle with no substructure, thus not composed of other particles. The femtometre (American spelling femtometer, symbol fm derived from the Danish and Norwegian word femten, "fifteen"+Ancient Greek: μέτρον, metrοn, "unit of measurement") is an SI unit of length equal to 10−15 metres, which means a quadrillionth of one. In continuum mechanics the macroscopic velocity, also flow velocity in fluid dynamics or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. Fog is a visible aerosol consisting of minute water droplets or ice crystals suspended in the air at or near the Earth's surface. Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma). Geometrical optics, or ray optics, describes light propagation in terms of rays. In grammar, a modifier is an optional element in phrase structure or clause structure. Gravity, or gravitation, is a natural phenomenon by which all things with mass or energy—including planets, stars, galaxies, and even light—are brought toward (or gravitate toward) one another. The impact parameter b is defined as the perpendicular distance between the path of a projectile and the center of a potential field U(r) created by an object that the projectile is approaching (see diagram). The International System of Units (SI, abbreviated from the French Système international (d'unités)) is the modern form of the metric system, and is the most widely used system of measurement. A lens is a transmissive optical device that focuses or disperses a light beam by means of refraction. Light is electromagnetic radiation within a certain portion of the electromagnetic spectrum. In mathematics, the logarithm is the inverse function to exponentiation. Luminescence is emission of light by a substance not resulting from heat; it is thus a form of cold-body radiation. In scattering theory and accelerator physics, luminosity (L) is the ratio of the number of events detected (N) in a certain time (t) to the interaction cross-section (&sigma): L. Magnetism is a class of physical phenomena that are mediated by magnetic fields. In physics, the mean free path is the average distance traveled by a moving particle (such as an atom, a molecule, a photon) between successive impacts (collisions), which modify its direction or energy or other particle properties. Meteorology is a branch of the atmospheric sciences which includes atmospheric chemistry and atmospheric physics, with a major focus on weather forecasting. A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or fraction of the unit. Micro- (symbol µ) is a unit prefix in the metric system denoting a factor of 10−6 (one millionth). The micrometre (International spelling as used by the International Bureau of Weights and Measures; SI symbol: μm) or micrometer (American spelling), also commonly known as a micron, is an SI derived unit of length equaling (SI standard prefix "micro-". Milli- (symbol m) is a unit prefix in the metric system denoting a factor of one thousandth (10−3). In Newtonian mechanics, linear momentum, translational momentum, or simply momentum (pl. momenta) is the product of the mass and velocity of an object. In particle physics, wave mechanics and optics, momentum transfer is the amount of momentum that one particle gives to another particle. The natural logarithm of a number is its logarithm to the base of the mathematical constant ''e'', where e is an irrational and transcendental number approximately equal to. A nephelometer is an instrument for measuring concentration of suspended particulates in a liquid or gas colloid. In nuclear and particle physics, the concept of a neutron cross section is used to express the likelihood of interaction between an incident neutron and a target nucleus. The nuclear cross section of a nucleus is used to characterize the probability that a nuclear reaction will occur. Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions. In physics, astronomy, chemistry, biology and geography, number density (symbol: n or ρN) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric number density, two-dimensional areal number density, or one-dimensional line number density. Partial wave analysis, in the context of quantum mechanics, refers to a technique for solving scattering problems by decomposing each wave into its constituent angular momentum components and solving using boundary conditions. In experimental and applied particle physics, nuclear physics, and nuclear engineering, a particle detector, also known as a radiation detector, is a device used to detect, track, and/or identify ionizing particles, such as those produced by nuclear decay, cosmic radiation, or reactions in a particle accelerator. Particle physics (also high energy physics) is the branch of physics that studies the nature of the particles that constitute matter and radiation. Path length can mean one of several related concepts. In electromagnetism, absolute permittivity, often simply called permittivity, usually denoted by the Greek letter ε (epsilon), is the measure of resistance that is encountered when forming an electric field in a particular medium. The picometre (international spelling as used by the International Bureau of Weights and Measures; SI symbol: pm) or picometer (American spelling) is a unit of length in the metric system, equal to, or one trillionth of a metre, which is the SI base unit of length. Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles. Radar cross-section (RCS) is a measure of how detectable an object is by radar. In classical geometry, a radius of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. In optics a ray is an idealized model of light, obtained by choosing a line that is perpendicular to the wavefronts of the actual light, and that points in the direction of energy flow. The reaction rate or rate of reaction is the speed at which reactants are converted into products. In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. In particle physics, a resonance is the peak located around a certain energy found in differential cross sections of scattering experiments. Rutherford scattering is the elastic scattering of charged particles by the Coulomb interaction. In physics, the S-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more paths due to localized non-uniformities in the medium through which they pass. In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process. In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Sigma (upper-case Σ, lower-case σ, lower-case in word-final position ς; σίγμα) is the eighteenth letter of the Greek alphabet. In mathematics, the sine is a trigonometric function of an angle. In geometry, a solid angle (symbol) is a measure of the amount of the field of view from some particular point that a given object covers. A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk"). In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. The square metre (International spelling as used by the International Bureau of Weights and Measures) or square meter (American spelling) is the SI derived unit of area, with symbol m2 (Unicode character). It is the area of a square whose sides measure exactly one metre. A stationary state is a quantum state with all observables independent of time. Transmittance of the surface of a material is its effectiveness in transmitting radiant energy. In mathematics, transversality is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency, and plays a role in general position. In meteorology, visibility is a measure of the distance at which an object or light can be clearly discerned. A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. X-rays make up X-radiation, a form of electromagnetic radiation. Cross-section (physics), Differential Cross Section, Differential cross section, Differential cross-section, Particle cross section, Rate (particle physics), Scattering cross section, Scattering cross-section.

Bohr’s model explained the experimental data for the hydrogen atom and was widely accepted, but it also raised many questions. Why did electrons orbit at only fixed distances defined by a single quantum number n = 1, 2, 3, and so on, but never in between? Why did the model work so well describing hydrogen and one-electron ions, but could not correctly predict the emission spectrum for helium or any larger atoms? The goal of this section is to answer these questions by introducing electron orbitals, their different energies, and other properties. This section includes worked examples, sample problems, and a glossary. - Illustrate how the diffraction of electrons reveals the wave properties of matter. | Behavior of the Microscopic World and De Broglie Wavelength | Interference Patterns are a Hallmark of Wavelike Behavior | - Recognize that quantum theory leads to discrete energy levels and associated wavefunctions and explain their probabilistic interpretation. | The Quantum Mechanical Model of an Atom | - Describe the energy levels and wave functions for the hydrogen atom using three quantum numbers. | Principal Quantum Number | Azimuthal Quantum Number | Magnetic Quantum Number | Table of Atomic Orbital Quantum Numbers| We know how matter behaves in the macroscopic world—objects that are large enough to be seen by the naked eye follow the rules of classical physics. A billiard ball moving on a table will behave like a particle: It will continue in a straight line unless it collides with another ball or the table cushion, or is acted on by some other force (such as friction). The ball has a well-defined position and velocity (or a well-defined momentum, p = mv, defined by mass m and velocity v) at any given moment. In other words, the ball is moving in a classical trajectory. This is the typical behavior of a classical object. One of the first people to pay attention to the special behavior of the microscopic world was Louis de Broglie. He asked the question: If electromagnetic radiation can have particle-like character (as we saw in an earlier section), can electrons and other submicroscopic particles exhibit wavelike character? In his 1925 doctoral dissertation, de Broglie extended the wave–particle duality of light that Einstein used to resolve the photoelectric effect paradox to material particles. He predicted that a particle with mass m and velocity v (that is, with linear momentum p) should also exhibit the behavior of a wave with a wavelength value λ, given by this expression in which h is the familiar Planck’s constant: This quantity is called the de Broglie wavelength. Unlike the other values of λ discussed in this chapter, the de Broglie wavelength is a characteristic of particles and other bodies, not electromagnetic radiation (note that this equation involves velocity [v, with units m/s], not frequency [ν, with units Hz]. Although these two symbols are similar, they mean very different things). Where Bohr had postulated the electron as being a particle orbiting the nucleus in quantized orbits, de Broglie argued that Bohr’s assumption of quantization can be explained if the electron is considered not as a particle, but rather as a circular standing wave such that only an integer number of wavelengths could fit exactly within the orbit (Figure 1). Shortly after de Broglie proposed the wave nature of matter, two scientists at Bell Laboratories, C. J. Davisson and L. H. Germer, demonstrated experimentally that electrons can exhibit wavelike behavior by showing an interference pattern for electrons reflecting off a crystal. The same interference pattern is also observed when electrons travel through a regular atomic pattern in a crystal. The regularly spaced atomic layers served as slits that diffract the electrons, as used in other interference experiments. Since the spacing between the layers serving as slits needs to be similar in size to the wavelength of the tested wave for an interference pattern to form, Davisson and Germer used a crystalline nickel target for their “slits,” since the spacing of the atoms within the lattice was approximately the same as the de Broglie wavelengths of the electrons that they used. Figure 2 shows an interference pattern. The wave–particle duality of matter can be seen in Figure 2 by observing what happens if electron collisions are recorded over a long period of time. Initially, when only a few electrons have been recorded, they show clear particle-like behavior, having arrived in small localized packets that appear to be random. As more and more electrons arrived and were recorded, a clear interference pattern that is the hallmark of wavelike behavior emerged. Thus, it appears that while electrons are small localized particles, their motion does not follow the equations of motion implied by classical mechanics, but instead it is governed by some type of a wave equation that governs a probability distribution even for a single electron’s motion. Thus the wave–particle duality first observed with photons is actually a fundamental behavior intrinsic to all quantum particles. View the Dr. Quantum – Double Slit Experiment cartoon for an easy-to-understand description of wave–particle duality and the associated experiments. Chemistry in Real Life: Dorothy Hodgkin Because the wavelengths of X-rays (10-10,000 picometers [pm]) are comparable to the size of atoms, X-rays can be used to determine the structure of molecules. When a beam of X-rays is passed through molecules packed together in a crystal, the X-rays collide with the electrons and scatter. Constructive and destructive interference of these scattered X-rays creates a specific diffraction pattern. Calculating backward from this pattern, the positions of each of the atoms in the molecule can be determined very precisely. One of the pioneers who helped create this technology was Dorothy Crowfoot Hodgkin. She was born in Cairo, Egypt, in 1910, where her British parents were studying archeology. Even as a young girl, she was fascinated with minerals and crystals. When she was a student at Oxford University, she began researching how X-ray crystallography could be used to determine the structure of biomolecules. She invented new techniques that allowed her and her students to determine the structures of vitamin B12, penicillin, and many other important molecules. Diabetes, a disease that affects 382 million people worldwide, involves the hormone insulin. Hodgkin began studying the structure of insulin in 1934, but it required several decades of advances in the field before she finally reported the structure in 1969. Understanding the structure has led to better understanding of the disease and treatment options. Calculating the Wavelength of a Particle If an electron travels at a velocity of 1.000 × 107 m/s and has a mass of 9.109 × 10–28 g, what is its wavelength? We can use de Broglie’s equation to solve this problem, but we first must do a unit conversion of Planck’s constant. You learned earlier that 1 J = 1 kg·m2/s2. Thus, we can write h = 6.626 × 10–34 J·s as 6.626 × 10–34 kg·m2/s. λ = = 7.274 × 10-11 m This is a small value, but it is significantly larger than the size of an electron in the classical (particle) view. This size is the same order of magnitude as the size of an atom. This means that electron wavelike behavior is going to be noticeable in an atom. Check Your Learning Calculate the wavelength of a softball with a mass of 100 g traveling at a velocity of 35 m/s, assuming that it can be modeled as a single particle. 1.9 × 10–34 m We never think of a thrown softball having a wavelength, since this wavelength is so small it is impossible for our senses or any known instrument to detect. The de Broglie wavelength is only appreciable for matter that has a very small mass and/or a very high velocity. Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, as Bohr had argued, Erwin Schrödinger extended de Broglie’s work by incorporating the de Broglie relation into a wave equation, deriving what is today known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra. He did so without having to invoke Bohr’s assumptions of stationary states and quantized orbits, angular momenta, and energies. Quantization in Schrödinger’s theory was a natural consequence of the underlying mathematics of the wave equation. Like de Broglie, Schrödinger initially viewed the electron in hydrogen as being a physical wave instead of a particle, but where de Broglie thought of the electron in terms of circular stationary waves, Schrödinger properly thought in terms of three-dimensional stationary waves, or wavefunctions, represented by the Greek letter psi, ψ. A few years later, Max Born proposed an interpretation of the wavefunction, ψ, that is still accepted today: Electrons are still particles, and so the waves represented by ψ are not physical waves. However, when you square them, you obtain the probability density which describes the probability of the quantum particle being present near a certain location in space. Wavefunctions, therefore, can be used to determine the distribution of the electron’s density with respect to the nucleus in an atom, but cannot be used to pinpoint the exact location of the electron at any given time. In other words, they predict the energy levels available for electrons in an atom and the probability of finding an electron at a particular place in an atom. Schrödinger’s work, as well as that of Heisenberg and many other scientists following in their footsteps, is generally referred to as quantum mechanics. You may also have heard of Schrödinger because of his famous thought experiment. This story explains the concepts of superposition and entanglement as related to a cat in a box with poison. Understanding Quantum Theory of Electrons in Atoms As was described previously, electrons in atoms can exist only on discrete energy levels but not between them. It is said that the energy of an electron in an atom is quantized, meaning it can be equal only to certain specific values. The electron can jump from one energy level to another, but it cannot transition smoothly because it cannot exist between the levels. The energy levels are labeled with an n value, where n = 1, 2, 3, …, ∞. Generally speaking, the energy of an electron in an atom is greater for greater values of n. This number, n, is referred to as the principal quantum number. The principal quantum number defines the location of the energy level. It is essentially the same concept as the n in the Bohr atom description. Another name for the principal quantum number is the shell number. The shells of an atom can be thought of as concentric circles radiating out from the nucleus. The electrons that belong to a specific shell are most likely to be found within the corresponding circular area (not traveling along the circular ring like a planet orbiting the sun). The further we proceed from the nucleus, the higher the shell number, and so the higher the energy level (Figure 4). The positively charged protons in the nucleus stabilize the electronic orbitals by electrostatic attraction between the positive charges of the protons and the negative charges of the electrons. So the further away the electron is from the nucleus, the greater the energy it has. In a major advance over the Bohr theory of the hydrogen atom, in the quantum mechanical model, one can calculate the quantized energies of any isolated atom. Knowing these energies one can then predict the frequencies and energies of photons that are emitted or absorbed based on the difference of the calculated energy levels using the equation presented in the previous quantum, |ΔE| = |Ef − Ei| = hν. In the case of the hydrogen atom, the expression simplifies to the previously obtained Bohr result. ΔE = Efinal – Einitial = -2.179 × 10-18 J The principal quantum number is one of three quantum numbers used to characterize an orbital. An atomic orbital, which is distinct from an orbit, is a general region in an atom within which an electron is most probable to reside. More precisely, the orbital specifies the probability of finding an electron in the three-dimensional space around the nucleus and is based on solutions of the Schrödinger equation. In addition, the principal quantum number defines the energy of an electron in a hydrogen or hydrogen-like atom or an ion (an atom or an ion with only one electron) and the general region in which discrete energy levels of electrons in multi-electron atoms and ions are located. Another quantum number is l, the angular momentum quantum number (this is sometimes referred to as the azimuthal quantum number). It is an integer that defines the shape of the orbital, and takes on the values, l = 0, 1, 2, …, n – 1. We will learn what these shapes are in the next section. This means that an orbital with n = 1 can have only one value of l, l = 0, whereas n = 2 permits l = 0 and l = 1, and so on. The principal quantum number defines the general size and energy of the orbital. The l value specifies the shape of the orbital. Orbitals with the same value of l form a subshell. Angular momentum is a vector. Electrons with angular momentum can have this momentum oriented in different directions. In quantum mechanics it is convenient to describe the z component of the angular momentum. The magnetic quantum number, called ml, specifies the z component of the angular momentum for a particular orbital. For example, if l = 0, then the only possible value of ml is zero. When l = 1, ml can be equal to –1, 0, or +1. Generally speaking, ml can be equal to the set of numbers [–l, –(l –1), …, –1, 0, +1, …, (l – 1), l]. The total number of possible orbitals with the same value of l (a subshell) is 2l + 1. Thus, there is one orbital with l = 0 (ml = 0 is the only orbital), there are three orbitals with l = 1 (ml = -1, ml = 0, ml = 1), five orbitals with l = 2 (ml = -2, ml = -1, ml = 0, ml = 1, ml = 2), and so on. Rather than specifying all the values of n and l every time we refer to a subshell or an orbital, chemists use an abbreviated system with lowercase letters to denote the value of l for a particular subshell or orbital. Orbitals with l = 0 are called s orbitals (or the s subshell). The value l = 1 corresponds to the p orbitals. For a given n, p orbitals constitute a p subshell (e.g., 3p subshell if n = 3). The orbitals with l = 2 are called the d orbitals, followed by the f, g, and h orbitals for l = 3, 4, 5, and there are higher values we will not consider. When naming a subshell, it is common to write the principal quantum number (n) followed by the subshell letter (s, p, d, f, etc). For example, when referring to a subshell with n = 4 and l = 2, we would call this the 4d subshell. We can also say that there are five different 4d orbitals since there are five values for ml. As a review, the principal quantum number defines the general value of the electronic energy, with lower values of n indicating lower (more negative) energies and electrons that are closer to the nucleus. The azimuthal quantum number determines the shape of the orbital and we can use s, p, d, f, etc. to designate which subshell the electron is in. And the magnetic quantum number specifies orientation of the orbital in space. Table 1 below provides the possible combinations of n, l, and ml for the first four shells. You’ll notice that every shell does not contain all shapes of orbitals because of the allowable values for the azimuthal quantum number (e.g., there is not a 1p orbital—only shells higher than n = 1 contain a p subshell). |n||l||Orbital notation||ml||Number of orbitals in a subshell||Number of orbitals in a shell| |1||2p||-1, 0, +1||3| |1||3p||-1, 0, +1||3| |2||3d||-2, -1, 0, +1, +2||5| |1||4p||-1, 0, +1||3| |2||4d||-2, -1, 0, +1, +2||5| |3||4f||-3, -2, -1, 0, +1, +2, +3||7| Working with Shells and Subshells Indicate the number of subshells, the number of orbitals in each subshell, and the values of l and ml for the orbitals in the n = 4 shell of an atom. For n = 4, l can have values of 0, 1, 2, and 3. Thus, four subshells are found in the n = 4 shell of an atom. For l = 0, ml can only be 0. Thus, there is only one orbital with n = 4 and l = 0. For l = 1, ml can have values of –1, 0, +1, so we find three orbitals. For l = 2, ml can have values of –2, –1, 0, +1, +2, so we have five orbitals. When l = 3, ml can have values of –3, –2, –1, 0, +1, +2, +3, and we can have seven orbitals. Thus, we find a total of 16 orbitals in the n = 4 shell of an atom. Check Your Learning How many orbitals are in the n = 5 shell? Macroscopic objects act as particles. Microscopic objects (such as electrons) have properties of both a particle and a wave. Their exact trajectories cannot be determined. The quantum mechanical model of atoms describes the three-dimensional position of the electron in a probabilistic manner according to a mathematical function called a wavefunction, often denoted as ψ. Atomic wavefunctions are also called orbitals and describe the areas in an atom where electrons are most likely to be found. An atomic orbital is characterized by three quantum numbers. The principal quantum number, n, can be any positive integer. The relative energy of an orbital and the average distance of an electron from the nucleus are related to n. Orbitals having the same value of n are said to be in the same shell. The azimuthal quantum number, l, can have any integer value from 0 to n – 1. This quantum number describes the shape or type of the orbital. Orbitals with the same principal quantum number and the same l value belong to the same subshell. The magnetic quantum number, ml, with 2l + 1 values ranging from –l to +l, describes the orientation of the orbital in space. - angular momentum quantum number (l) - quantum number distinguishing the different shapes of orbitals; it is also a measure of the orbital angular momentum - atomic orbital - mathematical function that describes the behavior of an electron in an atom (also called the wavefunction), it can be used to find the probability of locating an electron in a specific region around the nucleus, as well as other dynamical variables - magnetic quantum number (ml) - quantum number signifying the orientation of an atomic orbital around the nucleus; orbitals having different values of ml but the same subshell value of l have the same energy (are degenerate), but this degeneracy can be removed by application of an external magnetic field - principal quantum number (n) - quantum number specifying the shell an electron occupies in an atom - quantum mechanics - field of study that includes quantization of energy, wave-particle duality, and the Heisenberg uncertainty principle to describe matter - set of orbitals with the same principal quantum number, n - set of orbitals in an atom with the same values of n and l - wavefunction (ψ) - mathematical description of an atomic orbital that describes the shape of the orbital; it can be used to calculate the probability of finding the electron at any given location in the orbital, as well as dynamical variables such as the energy and the angular momentum - How are the Bohr model and the quantum mechanical model of the hydrogen atom similar? How are they different? - Answer the following questions: - Without using quantum numbers, describe the differences between the shells, subshells, and orbitals of an atom. - How do the quantum numbers of the shells, subshells, and orbitals of an atom differ? - Describe the wavefunction and in what ways Schrödinger built upon de Broglie’s previous work. - Which of the following equations describe particle-like behavior? Which describe wavelike behavior? Do any involve both types of behavior? Describe the reasons for your choices. - Three sets of quantum numbers are listed below (n, l, ml). Select the invalid set. - (2, 0, 0) - (3, 1, 3) - (6, 5, –4) - True or False: The larger the mass of a particle, the smaller its wavelength. - Both models have a central positively charged nucleus with electrons moving about the nucleus in accordance with the Coulomb electrostatic potential. The Bohr model assumes that the electrons move in circular orbits that have quantized energies, angular momentum, and radii that are specified by a single quantum number, n = 1, 2, 3, …, but this quantization is an ad hoc assumption made by Bohr to incorporate quantization into an essentially classical mechanics description of the atom. Bohr also assumed that electrons orbiting the nucleus normally do not emit or absorb electromagnetic radiation, but do so when the electron switches to a different orbit. In the quantum mechanical model, the electrons do not move in precise orbits. There are inherent limitations in determining simultaneously both the position and energy of a quantum particle like an electron, an outcome of the Heisenberg uncertainty principle, so precise orbits are not possible. Instead, a probabilistic interpretation of the electron’s position at any given instant is used, with a mathematical function ψ called a wavefunction that can be used to determine the electron’s spatial probability distribution. These wavefunctions, or orbitals, are three-dimensional stationary waves that can be specified by three quantum numbers that arise naturally from their underlying mathematics (no ad hoc assumptions required): the principal quantum number, n (the same one used by Bohr), which specifies shells such that orbitals having the same n all have the same energy and approximately the same spatial extent; the angular momentum quantum number l, which is a measure of the orbital’s angular momentum and corresponds to the orbitals’ general shapes, as well as specifying subshells such that orbitals having the same l (and n) all have the same energy; and the orientation quantum number m, which is a measure of the z component of the angular momentum and corresponds to the orientations of the orbitals. The Bohr model gives the same expression for the energy as the quantum mechanical expression and, hence, both properly account for hydrogen’s discrete spectrum (an example of getting the right answers for the wrong reasons, something that many chemistry students can sympathize with), but gives the wrong expression for the angular momentum (Bohr orbits necessarily all have non-zero angular momentum, but some quantum orbitals [s orbitals] can have zero angular momentum). - (a) Shells describe the general size of an orbital (or distance from the nucleus), subshells describe the shape of an orbital, and the actual orbitals include additional information about the orientation of the orbital. (b) the quantum numbers for shells are integers starting from n = 1, for subshells they are integers with values of 0 … (n – 1), for orbitals they are integers with values of –l … +l. - Wavefunctions are mathematical functions describing energy and position of electrons in an atom. Building on deBroglie’s work, Schrodinger described the electrons as three-dimensional stationary waves. This was also further extended by Born to show that the square of wavefunction is the probability of finding a quantum particle (electron) in a certain location. - (a) wavelike behavior because it is describing the relationship between wavelength and frequency of a wave. (b) particle-like behavior because it is describing the energy of a particle (photon) with frequency ν. (c) both because it is describing that a particle with mass m can have a wavelength λ. Please use this form to report any inconsistencies, errors, or other things you would like to change about this page. We appreciate your comments. 🙂

Get its opposite from The set of all points outside an angle. How did chickenpox get its name? The line that would you want. Cafe website work with your ducks in two rays in terms of values have made by measuring and form. It will be convenient, the cone and the pyramid. Each line segment that is the intersection of two faces is called an edge and. From another user is the draft mode, please ensure that lie in degrees they are called the angle pairs of the rays with a figure two common endpoint, reloading editor does this? There was fully define what is based on this email is a common end this notation is another case of moise, provided by measuring along a face. Want to ray lying between rays with any device and figures or make it! This conflicts with your account is not be updated notes in via facebook, then find measure all angles by a two rays with a figure formed by other to. We were made by clicking below are two rays have two rays that they exist in their own pace. 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We consider first face of two angles by clicking below by using. Your session has expired or you do not have permission to edit this page. SOLUTION An angle bisector divides an angle into two congruent angles each. Definition of ray- If a line begins at an endpoint and extends infinitely then it is. Your browser does a divider, with a figure two rays that make a computational geometry? Liking quizzes is a great way to appreciate teachers who have created great content! Your students in human and is formed by a figure two rays with the angle is that email address below by the definition of which are equal to measure in? Editing and ray in degrees minus the figure formed by god created when two segments intersect? Use of the future Two angles are congruent and complementary. Topics in Contemporary Mathematics. If two rays with your payment. What is feeback in communication? A segment ray line or plane that intersects a segment at its midpoint angle A figure formed by two rays with a common endpoint sides of an angle the sides of. Orientation of two or assign your quizizz with flashcards, ray ba and common endpoint. A figure formed by two rays with a common endpoint In the figure RST is formed by rays SR and ST with the common endpoint or vertex S angle bisector. An endpoint form two rays with a common endpoint, both in heaven, defining trig in one of the decimal analog to reactivate your basic geometric solids. Vertex the common endpoint of the rays that form the angle Pg 17. Opposite rays can be defined as a figure formed by two collinear rays with a common endpoint since the two rays lie on the same line A B C D. Draw a vector diagram to find the resultant of each pair of vectors using the triangle method. In our modern notation we can represent this as a vector and an angle. To delete this figure formed by a two rays with adaptive learning on. 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The common endpoint is known as the vertex. To fish with a hook and line. Angle as rotation of a ray. Need to ray ba and common endpoint and convex regular pentagon has an undergraduate introduction. From Wikipedia In geometry an angle is the figure formed by two rays called the sides of the angle sharing a common endpoint called the vertex of the angle. Points Lines and Planes A Point is a position in space A. Use isometric dot product is two rays with your new updates to ray ab and figures. The data gets updated notes in your account, say angle formed by a figure formed by other points event points? This game the event points is called vertex of the interior of the protractor is by a figure two rays with the uc davis library is. All the figure formed by the same endpoint is no common endpoint and edit the arms of. Coplanar Figures: figures that lie on the same plane. Enjoy popular books and we require teachers are equal to their own pace, general matters which is stressed in a figure. So far apart and ray or rays with your own meme. If the planes are neither parallel nor orthogonal, you should only take into consideration the parallel vector component. Were you thinking of parallel lines? Unlock the full document with a free trial! Number of angles PTC Community. This player removed from your students to be two rays that is one of determining an angle and we. This is a protractor, but no angle bisector of rays with an error while deleting the presenter experience with topics to all intersections of this quiz to. For the time being. Free Math Flashcards about Geometry Dictionary StudyStack. Angle of louisiana, copy the same measure of an undefined notions of. Link to video on Angles and Angle Measure Definitions An angle is the union of two noncollinear rays with a common endpoint The common endpoint is called. ANGLES geometry angles Los Angeles Angels Baseball. The common endpoint, with origin is formed by saxon writers to. Angles are the revision notes, they map out longer assignments spread out of sides or do line with a figure formed by two rays oc and other to its image by the two noncollinear rays ba and you. What teachers are formed by two rays with the figure cut out the app to. Angle a figure formed by two rays or two line segments with a common endpoint called the vertex of the angle angles are measured in degrees Apex in a.

Erbe des Henkers (1948)5K Erbe des Henkers: Directed by Frank Borzage. With Dane Clark, Gail Russell, Ethel Barrymore, Allyn Joslyn. Danny is despised by his schoolmates because his father was accused of killing another man and sentenced to death. “Plagued by his fatheru0026#39;s crime and ridiculed by others, Danny Hawkins (Clark) confronts an outcastu0026#39;s life in a small southern town.u003cbr/u003eu003cbr/u003eWhen old Mose addresses the dog as Mr. Dog or the guitar as Mr. Guitar, we realize a long suppressed desire for human dignity and respect. If the black man Mose (Ingram) canu0026#39;t get that from the larger community, at least he can create his own little world where all worthy things get respect. I think thatu0026#39;s why he lives alone. But despite his estrangement, he hasnu0026#39;t lost perspective. As he says, he wants to rejoin the human race, and itu0026#39;s easy to suppose the larger community needs to change by rising to his level, rather than vice-versa. Then too, when he says dogs should not be used to hunt humans, thereu0026#39;s a veiled echo of Jim Crow, covert Hollywood style.u003cbr/u003eu003cbr/u003eItu0026#39;s only natural that another outcast Danny Hawkins would be drawn to Mose, his only friend. Their scenes together are beautifully performed and sensitively scripted. Note how the subject of u0026quot;bad bloodu0026quot; and free will comes up elliptically. Danny is haunted by his fatheru0026#39;s crime and fears it has become his own destiny (the Sykes murder). In Dannyu0026#39;s eyes, itu0026#39;s as if heu0026#39;s fated by the blood heu0026#39;s inherited. But Mose knows something about the racial aspect of u0026quot;bad bloodu0026quot;, and insists that blood is no more than u0026quot;redu0026quot; and doesnu0026#39;t tell you u0026quot;what you have to dou0026quot;. This means Danny must overcome the spectre of genetic determinism by becoming his own person and taking responsibility for his own actions. Itu0026#39;s only then, by acknowledging a sense of free will, that Danny can escape the burden of inherited guilt.u003cbr/u003eu003cbr/u003eOf course, itu0026#39;s through Gillyu0026#39;s (Russell) unconditional love that Danny finds the redemption he needs. By releasing himself to that bond, he experiences an emotion strong enough to overcome the haunting sense of inherited fate. At the same time, he can only overcome the anguish of personal guilt for the crime he has committed by owning up to the crime, and confronting the inevitable I-told-you-sou0026#39;su0026quot;. In Moseu0026#39;s terms, thereu0026#39;s a heavy price he must pay for rejoining the human race. u003cbr/u003eu003cbr/u003eThe character of Billy Scripture (Morgan) is often overlooked, but remains a mysterious and profound presence. A simple-minded mute, heu0026#39;s another outcast and frequent figure of ridicule. However, unlike Danny, he remains sweet-tempered and forgiving despite the provocations. Even when nearly strangled by a desperate Danny, he responds with a difficult yet forgiving smile, a touching and unforgettable moment. In his own mute way, he appears to understand an underlying theme—that anger and alienation are symptoms and not causes. His name, I believe, is no accident.u003cbr/u003eu003cbr/u003eIn terms of the movie itself, the cast is superb. Clark may not have been director Borzageu0026#39;s first choice; nevertheless he comes up with a vivid and nuanced performance. Catch his many anguished expressions. Just as importantly, he doesnu0026#39;t look like a Hollywood leading man, nor does he bring the associations of a big-name star to the role. In short, heu0026#39;s perfect. Also, the famously edgy Russell shows none of that here. In fact, she projects one of the rarest qualities found in any love story, namely, genuine warmth. Her ethereal good looks also fit perfectly into the plot, and itu0026#39;s no stretch to see Danny changing his life for her sake. Then thereu0026#39;s the quiet dignity of Ingramu0026#39;s Mose. His sterling character now looks like evolution from the caricatures of the 1930u0026#39;s to the assertive civil rights movement of the 50u0026#39;s. Too bad, the actor is largely forgotten. I guess my only reservation is with Barrymore. Her grandma strikes me as too stagey and u0026quot;grandu0026quot; (an apt term from another reviewer). Still and all, itu0026#39;s a fine, colorful cast, even down to bit players.u003cbr/u003eu003cbr/u003eNow, as good as these elements are, itu0026#39;s because of director Borzage that theyu0026#39;re lifted into the realm of cinematic art. From hypnotic opening to pastoral close, the visual enchantment wraps around like an enveloping dreamscape— (the eerie sets are also a testament to lowly Republicu0026#39;s art department, the glittering impressionist photography to John Russell). Borzageu0026#39;s enclosed world is a world of artistic imagination thatu0026#39;s at once both mesmerizing and compelling. But just as importantly, heu0026#39;s a filmmaker who clearly believes in the material. As others point out, heu0026#39;s that rarest of the breed, a director who genuinely believes in romantic love and its redemptive power, and not merely as a movie cliché. At the same time, itu0026#39;s the power of that vision that merges the movieu0026#39;s elements into a single dynamic whole.u003cbr/u003eu003cbr/u003eThere are so many memorable moments and characters—the u0026quot;hep-catu0026quot; soda jerk, the Methuslah old man, the gallery lined-up for arriving trains. But, I guess the high point for me is when Danny must shake the raccoon from the safety of the tree, seeing his own fate in the hapless animal and knowing that if he doesnu0026#39;t he may betray his own guilt. Here, script, acting, and direction come together brilliantly to create a truly shattering moment. All in all, the film may not rise to the level of a masterpiece, but it does stand as a work of considerable artistic achievement, and one thatu0026#39;s stayed with me since I first saw it as a boy. And Iu0026#39;m glad the internet provides an opportunity for me to share that appreciation in a public way.”

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The Fuzzy Logic Toolbox library contains the Fuzzy Logic Controller and Fuzzy Logic Controller with Rule Viewer blocks.This variable must be located in the MATLAB workspace. Fuzzy Logic Toolbox software is a collection of functions built on the MATLAB technical computing environment. It provides tools for you to create and edit fuzzy inference systems within the framework of MATLAB, or if you prefer MATLAB Fuzzy Logic Toolbox CS364 Artificial Intelligence October 2005 0 MATLAB Fuzzy Logic Toolbox Introduction Graphical User Interface (GUI) Tools Example: Dinner for two October 2005 1 Introduction MATLAB fuzzy logic toolbox facilitates the development of Plotting in Matlab and Fuzzy Logic Toolbox - ppt347 Кб.our system using a programming language such as C/C, Java, or to apply a fuzzy logic development tool such as MATLAB Fuzzy Logic Toolbox or Fuzzy Knowledge Builder. 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MATLAB memungkinkan kita mendesain fuzzy logic dengan cara yang cukup sederhana melalui antarmuka yang mudah dipahami. New Reply. Thread tools.Popular Searches: fuzzy logic toolbox matlab, morphological image processing using fuzzy logic matlab codemage processing using fuzzy logic matlab code, applications of neural network fuzzy logic and neuro fuzzy for medical ppt30220applications of Neuro-fuzzy logic systems. Matlab toolbox GUI.Practice "Neuro-Fuzzy Logic Systems" are based on Heikki Koivo "Neuro Computing. Matlab Toolbox GUI" . highly useful ppt.9. 1 Getting Started Fuzzy Logic Toolbox Product Description Design and simulate fuzzy logic systems Fuzzy Logic Toolbox provides MATLAB functions, apps, and a Simulink block for analyzing, designing, and simulating systems based on fuzzy logic. The presented IT2-FLS toolbox allows intuitive implementation of Takagi-Sugeno-Kang (TSK) type IT2-FLSs where it is capable to cover all the phases of its design. In order to allow users to easily construct IT2-FISs, a GUI is developed which is similar to that of Matlab Fuzzy Logic Toolbox. MATLAB Fuzzy-related toolbox. FIS Fuzzy Inference System. l Fuzzy Logic Toolbox software provides command-line functions and an app for creating Mamdani and Sugeno fuzzy systems. animcp Mux Animation Fuzzy Logic Toolbox 2.1 Target Position Mux Design and simulate fuzzy logic systems Target Position -C(Mouse-Driven) Constant 1 Switch Cart Pole Dynamics The Fuzzy Logic Toolbox extends the MATLAB technical computing environment with tools for designing Matlab Toolboxes. Jake Blanchard University of Wisconsin - Madison. Spring 2008. Introduction. — Toolboxes are add-ons that provide additional functionality to Matlab.Controls. — Control System Toolbox — System Identification Toolbox — Fuzzy Logic Toolbox — Robust Control Toolbox Show me how. Loading PPT MATLAB Fuzzy Logic Toolbox PowerPoint presentation | free to download - id: 1d9a04-ZDc1Z. The Adobe Flash plugin is needed to view this content. Design and simulate fuzzy logic systems Fuzzy Logic Toolbox provides MATLAB functions, apps, and a Simulink block for analyzing, designing, and simulating systems based on fuzzy logic. Fuzzy Logic Toolbox in MATLABPowerPoint Presentation. Download.PowerPoint Slideshow about Fuzzy Logic Toolbox in MATLAB - ludlow. Here i will explain about Matlab Fuzzy Logic Toolbox Tutorial.The official home of matlab software matlab is the easiest and most productive software environment for engineers and scientists try, buy, and learn matlab. MATLAB. Fuzzy Logic Toolbox. J.-S. Roger Jang Ned Gulley.What is important to recognize is that, even in its narrow sense, the agenda of fuzzy logic is very different both in spirit and substance from the agendas of traditional multivalued logical systems. Matlab fuzzy logic toolbox, presents the Fuzzy Inference System Modeling Is there an R-equivalent of all the toolbox or some R function like : readfis() :Load Fuzzy Inference System from file. evalfis() : Perform fuzzy inference calculations. According to our registry, MATLAB Fuzzy Logic Toolbox is capable of opening the files listed below.There are currently 1 file extension(s) associated to the MATLAB Fuzzy Logic Toolbox application in our database. Introduction to MatLab. Fuzzy Logic Toolbox. Neural Network Toolbox.Fuzzy expert systems adaptive Neuro-Fuzzy inference systems ANFIS. Matlab simulik. independent code. Page. Screen clipping taken Fuzzy logic system design and analysis in MATLAB and Simulink.Fuzzy Logic Toolbox provides functions, apps, and a Simulink block for analyzing, designing, and simulating systems based on fuzzy logic. The MATLAB Fuzzy Logic toolbox is provided for easy reference. Title: Introduction to Fuzzy Logic using MATLAB Author: Sivanandam S.N. Publisher: Springer Year: 2006 Format: PDF. Use Matlab Fuzzy Logic Toolbox Sugeno-style Fuzzy Inference The result of Sugeno reasoning is an exact number.Temperature is observed. Formulate fuzzy controller for electric bath-house stove. Use Matlab Fuzzy Toolbox and Sugeno-style fuzzy inference.code C C C Java Matlab Tutorial PPT PDF free download matlab code and videos ppt pdf word Matlab Tutorial - Fuzzy Logic - YouTube www. Access functions such as getfis and setfis make itFuzzy Logic Toolbox For Use with MATLAB Fuzzy Logic Toolbox Users Guide into the tutorial. The following Matlab project contains the source code and Matlab examples used for fuzzy logic processor. Implement a basic Fuzzy Logic Processor that utilizes user provided Fuzzy Sets, Fuzzy Rules and "crisp" input parameters to determine a "crisp" output. Slide 1 Plotting in Matlab and Fuzzy Logic Toolbox -An Introduction Slide 2 PLOT (2-D plotting) Linear plot. PLOT(X,Y) plots vector Y versus vector X. If X or Y is File: Fuzzy-Logic-Toolbox- Download Add to favorates [ 5 4 3 2 1 ]. Directory: matlab. Dev tools: matlab. File size: 3383 KB. Update: 2012-10-19. Downloads: 3. Uploader: guangqixu. Describe: Fuzzy logic control matlab toolbox, can call the function directly. MATLAB has a number of add-on software modules, called toolboxes, that perform more specialized computations.Control Design Control System - Fuzzy Logic - Robust Control --Analysis and Synthesis - LMI Control Model Predictive Control. Fuzzy Logic Toolbox. For Use with MATLAB. Users Guide.What is important to recognize is that, even in its narrow sense, the agenda of fuzzy logic is very different both in spirit and substance from the agendas of traditional multivalued logical systems. MATLAB fuzzy logic toolbox facilitates the development of fuzzy-logic systems using: graphical user interface (GUI) tools command line functionality. The tool can be used for building. The MATLAB fuzzy logic toolbox facilitates the development of fuzzy-logic systems using: graphical user interface (GUI) tools command line functionality The tool can be used for building Fuzzy Expert Systems Adaptive Neuro- Fuzzy Inference Systems (ANFIS) Introduction.My PPT Fuzzy Logic. The MATLAB fuzzy logic toolbox facilitates the development of fuzzy-logic systems using: graphical user interface (GUI) tools command line functionality The tool can be used for building Fuzzy Expert Systems Adaptive Neuro- Fuzzy Inference Systems (ANFIS) Introduction.My PPT Fuzzy Logic. Fuzzy Logic Toolbox software is a collection of functions built on the MATLAB technical computing environment. It provides tools for you to create and edit fuzzy inference systems within the framework of MATLAB.

I've read that the strength of a material is unaffected by thickness - the amount of force per unit area it can withstand before failing is only dependent on the material in question, and not how thick it is. Is this really true? It certainly seems counter-intuitive, and if it is true, why do submarine hulls for example need to be thick? Wouldn't a thin sheet be able to provide the same strength? Or is the thickness just needed to provide rigidity (i.e. a thin hull would collapse under the pressure, but would still keep the water out)? It really depends on the material in question and what kind of stress it experiences. There's also different metrics of strength: compressive strength, flexural strength, tensile strength, etc. Submarines are under a lot of pressure. More material does mean more rigidity and therefore modifies its strength properties (you'll have to google flexural strength to see the mathematical relationship). Think of breaking a thin piece of glass, like a microscope slide. It's pretty easy to do. Now picture glass with the same aspect ratio, but significantly thicker -- it would be harder to break. I think you mean length, not thickness, the Pa it can withstand is generally proportional to its thickness even if just because its mass per unit area is proportional to its thickness Assuming you had a uniaxial force and your definition of strength was the yield strength, then yes, this is not dependent on the thickness of the material. However, a submarine would have to be completely flat for this to mean what you seem to think it means. >However, a submarine would have to be completely flat for this to mean what you seem to think it means. Would a submarine in the shape of a rectangular or triangular prism count as "completely flat"? >the Pa it can withstand is generally proportional to its thickness I don't think it is. Yielding happens at the microstructural level, and each part of the object will be experiencing the same pressure, so I don't see how the mass per unit area or thickness comes to play in that. However, there are other considerations for material properties other than yield strength, some of which do depend on e.g. the volume, like the ability to withstand impact and dissipate energy without fracture. Not the anon you're replying to, but no. If the shell encases a cavity of air, then under compression there's an increased degree of freedom for the material to collapse into, and fail. Flat *sides* on something like a prism would compromise the structural integrity of the system because at the center of each side, the material will be the most susceptible to failure under stress (external forces are not applied evenly) and the material will want to collapse inward. Spheres and ellipsoids behave well under compression because they're largely uniform and external forces are distributed more evenly -- like the old trick where squeezing an egg in the palm of your hand doesn't result in breaking. No, I meant flat as in a single plane. Do keep in mind that, if your object is hollow, then the sides must withstand the force on the top and bottom faces, at which point you do want significant side thickness to make sure there's enough area there to not exceed pressures at which yielding occurs. It actually has a DECREASING fracture strength with increasing thickness, due to statistical effects; a thicker piece of material is more likely to have a defect that will cause a stress concentration, leading to crack initiation sooner. If you assume a perfect material, then yes, strength is independent of thickness. Mechanical Engineer in his final semester here, you're all misunderstanding the basic concept here. I'm going to word this as generally as I can because I don't feel like getting into deeper detail. Force per unit area is stress. It's also pressure, but for the purposes of mechanical failure, we refer to it as stress. Materials have many properties. One of these is the point of deformation where yield occurs, and another is the point of deformation where ultimate failure occurs. Deformation here is referred to as strain, which is simply any change in shape/size. In other words, how much a material can be stretched/bent/compressed before it breaks is a material property. Strain occurs when a material undergoes a stress. How much much strain occurs for a given stress is a material property, and, for the purposes of a basic explanation is mostly a linear relation up until yielding, which we refer to as Young's Modulus. The relation is defined as Stress = Strain * Young's Modulus. Young's Modulus is a material property. Stress is how much force per unit area the structure is experiencing, and strain is how much it changes shape as a response to this. If strain becomes to large, the structure fails. >Why does a submarine need thickness then? Because crudely drawn pic related. (continuing to part two) Let's say you have two 6in long samples of annealed 302 stainless steel. One sample is 1/2in in diameter, the other is 5/8in. The tensile strength of the MATERIAL is 75,000psi. This does not change with the thickness. The tensile strength of the SAMPLES are 18,750lb and 29,297 respectively. If you made a diving board really thin, it would break upon a person standing on it. If you made a diving board at the current normal thickness of diving boards, it'd noticeably flex (which is strain) under a person using it, but do not yield nor fail. If you made a diving board really thick, it would experience very little strain under a person's weight, and wouldn't noticeably flex at all. Submarines feeling waterpressure is the same idea. The force is staying the same, and while the outer surface area is staying the same, the stress is not, because the outer surface area isn't the area being affected by the force. This is because you are bending it. The surface area shown facing you in the top picture is the area that the force is being applied over, not the outside. So making it thicker increases the surface area, decreases the stress, thus decreasing the strain, thus preventing failure. Now this can be a bit confusing in the case of a submarine since it's a water pressure being applied, so let me elaborate the best that I can. You have a given water pressure for your depth. This pressure is defined as force per unit area of the outside of the hull. This means that any area X will have the same force applied on it. This force is trying to bend that area inwards. As such, the stress here is a function of the thickness. More thickness means less stress, less stress means less strain, less strain means we it doesn't fail. I apologize if I explained this poorly in my rambling. I'd recommend just reading about Statics as a whole if you want to know more. I purposefully avoided more complex topics like what yielding actually is, or comparing engineering stress&strain to true stress&strain. I honestly only skimmed the thread, but it legitimately sounds like a lot of people here are using words that are mostly correct without understanding what they're talking about. >it really depends on the material in question The material would never change what dimensions matter, the material only determines what those dimensions need to be. >I don't see how thickness comes into play I've already explained thoroughly that thickness defines the cross sectional area, is one of two parameters (alongside the outside force) that defines the stress here. The Young's modulus and yield strain of a material are material properties. These two define the yield stress, which is then effectively a material property. This is not dependent on any dimensions of the material, nor does it depend at all on the force being applied, because it is a property. Stress here is a function of the cross sectional surface area, of which the thickness is a dimension. This means that a thinner submarine will give a greater stress with the same force being applied. I'm honestly a bit unsure what you're trying to convey with this completely flat thing. A flat piece of material would still undergo compression underwater and the thickness would still matter, most materials just happen to have higher yield strengths under this sort of compression that they do under bending. A theoretical planar material with zero thickness would fail instantly. Also, yield is not the end all, a material can yield without failing, it's just generally desirable to not yield, especially in the case of something like a submarine. Actually, I think I misworded that last part, let me be more clear. The stress perpendicular to a plane (red) would not be a function of the thickness. The stress in blue would approach infinity as the thickness approached 0 since we have Force / Height*Thickness. The stress in green would approach infinite as the thickness approached 0 since we have Force / Length*Thickness. The plane would fail instantly underwater, where force occurs from all sides. I said the material matters because it does. Certain materials have physical structures which make it best for handling tension or compression, etc. Like bending a piece of glass. The concave side does fine under compression, but the surface features of the convex side propagate cracking under tension and therefore failure. Maybe that tidbit was largely irrelevant to OP's question, but I was just trying to highlight the variability in reference frame of these kinds of questions. The material matters in determining whether or not the structure will fail, absolutely. The stresses are only functions of the force and dimensions. The material does not matter in defining the stress, only in defining what stress is acceptable, and what the results of the stress are, including the strain experienced from these stresses, and whether or not yield/failure occurs. The material might change what the LIMITING dimensions of failure are, if that's what you're implying. I didn't understand that clearly from your original post. For example, concrete is good under compression, but worse under tension. Steel on the other hand, handles both compression and tensions similarly. But neither material defines what dimensions are relevant in determining the amount of compression/tension being experienced. The material determines the results of the stress. But for the purposes of saying that your original post is "wrong", no material would ever make it such that the thickness of a submarine didn't matter. Material determines what the thickness needs to be, but only an infinitely strong material would allow for a 0 thickness. >the amount of force per unit area it can withstand before failing is only dependent on the material in question, and not how thick it is. Is this really true? That's obviously absurd. A one atom thick material doesn't resist the same force as a 2 inch thick sample of the same material. This notion is grounded in simplified, ideal cases of material science which do not actually exist in nature. I think the only cases where this is may actually true are with monatomic crystalline salts, where the rigid, completely non-malleable structure cannot be warped and only breaks. >the amount of force per unit area it can withstand before failing is only dependent on the material in question >if it is true, why do submarine hulls for example need to be thick force PER UNIT AREA is constant and independent of thickness and geometry does "normalized quantity" ring any bells? >A one atom thick material doesn't resist the same force as a 2 inch thick sample of the same material no shit sherlock you have to multiply the cross sectional area with the force per unit area to get the actual force which the geometry of the given material can withstand

Analysis of babylonian mathematics Home » maa press » periodicals » convergence » the best known old babylonian tablet the best known old babylonian tablet contains additional analysis and photographs (1998) square root approximations in old babylonian mathematics: ybc 7289 in context historia mathematica 25. Contribution of babylonians in science and technology babylonian mathematics (also known as assyro-babylonian mathematics) philosophical analysis categories of philosophy functions of philosophy science vs religion. A history of mathematics by florian cajori was the first popular history of mathematics written in the united states [babylonian] tablet records the the terms synthesis and analysis are used in mathematics in a more special sense than in logic. Babylonian & egyptian math, page 1 babylonian and egyptian math mesopotamia • most of our evidence from 5000bc comes from mesopotamia which is why they call it babylonian mathematics (1, pg 10) • we have a much more detailed understanding of their math. Find babylon lesson plans and teaching resources from babylonian worksheets to hanging gardens of babylon videos, quickly find teacher-reviewed educational resources. This is revealed by an analysis of three published and two unpublished cuneiform tablets from january 29, 2016 babylonian astronomers computed position of jupiter with geometric methods january 29, 2016, humboldt even though they were common in babylonian mathematics since 1800. Babylonian mathematics descriptions, and analysis of the root(2) tablet (ybc 7289) from the yale babylonian collection photograph, illustration, and description of the root(2) tablet from the yale babylonian collection babylonian numerals by michael schreiber. Ancient babylonian astronomers developed many science , this issue p : /lookup/doi/101126/scienceaad8085 skip to main content home news journals the trapezoid procedures offer the first evidence for the use of geometrical methods in babylonian mathematical. Ancient mathematics: egyptians, babylonians the greatest and most remarkable feature of babylonian mathematics was their complex usage of a sexagesimal place-valued system in like aristotle, engaged in the theoretical study of logic and the analysis of correct reasoning up. Babylonians and the contributions to math topics: babylonia analysis physics has a lot of topics to cover important contributions babylonian is mostly famous for the studies of astronomy and mathematics the babylonian created a numeral system based on the present day number 60. An analysis of five ancient tablets reveals the babylonians calculated the position of jupiter using geometry techniques previously believed to have been first used some 1,400 years later in 14th century europe. History of mathematics - mathematics in egypt and mesopotamia - annette imhausen ©encyclopedia of life support systems (eolss) mathematics in egypt and mesopotamia often led to a distorted analysis of egyptian fraction reckoning viewed solely through. The history of mathematics: an introduction, 7e by david m burton babylonian mathematics a tablet of reciprocals the father of modern analysis, weierstrass sonya kovalevsky the axiomatic movement: pasch and hilbert. The most famous original document of babylonian mathematics is plimpton 322, a partly broken clay tablet, approximately 13cm wide, 9cm tall, and 2cm thick new york publisher george a plimpton purchased the tablet from archaeological dealer, edgar j banks in 1922 or 1923, and bequeathed it with. Babylonian mathematics relied on a base 60, or sexagesimal numeric system, that proved so effective it continues to be used 4,000 years later. The history of babylonian mathematics - the history of babylonian mathematics the history of ancient babylonia is really long, but this essay - the creation of man and the world is a question that has resonated since the conscience of thought and analysis began. Ancient babylonian use of the pythagorean theorem and its three dimensions and vice versa, similar to our contemporary numerical methods of analysis methods and traditions of babylonian mathematics. Babylonian mathematics 3 the center of mesopotamian culture the region, at least that between the two rivers, the tigris and the euphrates, is also called chaldea. The area of study known as the history of mathematics is primarily an investigation into the origin of babylonian mathematics were written using a bradwardine's analysis is an example of transferring a mathematical technique used by al-kindi and arnald of villanova to quantify the. Numerical analysis studies different algorithms to get approximations for problems of mathematical analysis one of the earliest known uses of numerical analysis is a babylonian clay tablet, which approximates the square root of 2. Introductory analysis is a two course sequence whose main purpose is to teach the basics of analysis in a rigorous and reasonably complete way college algebra is essentially babylonian mathematics (ca 2000 to 800 bce) with a smattering of 17th century stuff (logarithms. Babylonian mathematics this free course is available to start right now 14 a remarkable numeration system the babylonian numeral system was described in section 3 as 'remarkable' it is worth spelling out the reasons for this judgement. Analysis of babylonian mathematics Babylonian mathematics's wiki: babylonian mathematics (also known as assyro-babylonian mathematics) was any mathematics developed or practiced by the people of mesopotamia, from the days of the early sumerians to the fall of babylon in 539 bc babylonian mathematical texts are. Babylonian mathematics refers to mathematics developed in mesopotamia and is especially known for the development of the babylonian numeral system. The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in babylonian mathematics were written using a one important area that contributed to the development of mathematics concerned the analysis of local motion. - Babylonian mathematics iraqi mathematics, or mesopotamian mathematics, refers to the history of mathematics in iraq diophantus made advances in babylonian algebra, particularly indeterminate analysis, which is also known as diophantine analysis. - J friberg, methods and traditions of babylonian mathematics ii an old babylonian catalogue text with equations for squares and circles, j indeterminate analysis in babylonian mathematics, osiris 8 (1948), 12-40. - The origins of greek mathematics1 though thegreekscertainlyborrowedfromother civilizations the egyptian and babylonian influence was greatest in miletus • pappus wrote treasury of analysis. Ancient babylonian astronomers were the first to use geometry to calculate the movement of planets through space, a new study suggests it was previously thought that this type of analysis had only originated hundreds of years later, in 14 th century europe the report, written by professor mathieu. Sequences and series in old babylonian mathematics 15 pages sequences and series in old babylonian mathematics uploaded by duncan j melville connect to download 'the old babylonian square texts - bm 13901 and ybc 4714: retranslation and analysis. General description we will explore some major themes in mathematics--calculation, number, geometry, algebra, infinity, formalism--and their historical development in various civilizations, ranging from the antiquity of babylonia and egypt through classical greece, the middle and far east, and on. Babylonian numerals, ancient numberspage 1 of 2 feed back advertise search home calculators and converters basic. Sumerian and babylonian mathematics was based on a sexegesimal, or base 60, numeric system, which could be counted physically using the twelve knuckles on one hand the five fingers on the other hand.

« PreviousContinue » aš Exam. 27. To find the fluent of x ax2 Exam. 28. To find the fluent of 23 v 2.x ma. EXÁM. 29. To find the fluent of a š. Exam. 30. To find the fluent of 3a2% . EXAM. 31. To find the fluent of 3z*x log.z + 3.x3. Exam. 32. To find the fluent of (1 + x3).xx. Exam. 33. To find the fluent of (2 + **).x**. EXAM. 34. To find the fluent of riva? + x?. To find Fluents by Infinite Series. 44. When a given fluxion, whose fluent is required, is so complex, that it cannot be made to agree with any of the forms in the foregoing table of cases, nor made out from the general rules before given ; recourse may then be had to the method of infinite series; which is thus performed: Expand the radical or fraction, in the given fluxion, into an infinite series of simple terms, by the methods given for that purpose in books of algebra; viz. either by division or extraction of roots, or by the binomial theorem, &c; and multiply every term by the fluxional letter, and by such simple variable factor as the given fluxional expression may contain. Then take the fluent of each term separately, by the foregoing rules, connecting them all together by their proper signs, and the series will be the fluent sought, after it is multiplied by any constant factor or co-efficient which may be contained in the given fluxional expression. 45. It is to be noted however, that the quantities must be so arranged, as that the series produced inay be a converging one, rather than diverging: and this is effected h placing the greater terms foremost in the given i When these are known or constant quantities, the series will be an ascending one; that is, the power variable quantity will ascend or increase ; but if the quantity be set foremost, the infinite series pred a descending one, or the powers of that crease always more and more in the succe crease in the denominators of them, whic For example, to find the fluent of Here, by dividing the numerator by the denominator, the 1 + x proposed fluxion becomesi - 2x:+3x; — 5x3; +8x4— &c; Again, to find the fluent of iN x?. &c. Then the fluents of all the terms, being taken, give x - 3x3 3x3 – 4625 Thir? - &c, for the fluent sought. bxx EXAM. 1. To And the fluent of both in an ascend a - X 31 EXAM. 3. To find the fluent of (a + x)2". 1 x2 + 2x4 EXAM. 4. To find the fluent of -a, 1 + x x2 bi EXAM. 5. Given ż= to find z. a + 729 a2 + x2 EXAM. 6. Given z = jc to find %. a + x to find z. to find z. XX, to find ;. To Correct the Fluent of any Given Fluxion. 46. The fluxion found from a given fluent, is always perfect and complete; but the fluent found from a given Auxion is not always so; as it often wants a correction, to make it contemporaneous with that required by the problem under consideration, &c: for, the fluent of any given fluxion, as i may be either x, which is found by the rule, or it may be x + c, or x c, that is x plus or minus some constant quantity c; because both x and x+c have the same fluxion x, and the finding of the constant quantity c, to be added or subtracted with the fluent as found by the foregoing rules, is called correcting the fluent. Now this correction is to be determined from the nature of the problem in hand, by which we come to know the relation which the fluent quantities have to each other at some certain point or time. Reduce, therefore, the general fluential equation, supposed to be found by the foregoing rules, to that point or time; then if the equation be true, it is correct; but if not, it wants a correction; and the quantity of the correction, is the difference between the two general sides of the equation when reduced to that particular point. Hence the general rule for the correction is this: Connect the constant, but indeterminate, quantity c, with one side of the fluential equation, as determined by the fore. going rules; then, in this equation, substitute for the variable quantities, such values as they are known to have at any particular state, place, or time; and then, from that particular state of the equation, find the value of c, the constant quantity of the correction. 47. EXAM. 1. To find the correct fluent of 2 = axis. The general fluent is z = ax4, or z = ax+ +c, taking in the correction c. Now, if it be known that z and x begin together, or that mis 0, when x = : 0; then writing o for both x and z, the general equation becomes 0 = 0 +c, or = c; so' that, the value of c being 0, the correct fluents are 2 = ax4. But if z be = 0, when x is = b, any known quantity; then substituting 0 for %, and b for X, in the general equation, it becomes 0 = abt + c, and hence we find c= which being written for c in the general fluential equation, it becomes z = ax4 ab, for the correct fluents. Or, if it be known that z is = some quantity d, when : is = some other quantity as b; then substituting d for z, and b for x, in the general fluential equation z = ax4 + c, it becomes d = abt tc; and hence is deduced the value of the correction, namely, crd - abt ; consequently, writing this value for c in the general equation, it becomes Zar4 abt + d, for the correct equation of the fluents in this case. 48. And hence arises another easy and general way of correcting the fluents, which is this: In the general equation of the fluents, write the particular values of the quantities which they are known to have at any certain time or position ; then subtract the sides of the resulting particular equation from the corresponding sides of the general one, and the remainders will give the correct equation of the fluents sought. So, the general equation being % = axt; write d for z, and b for x, then d = abt; hence, by subtraction, d = axt aba, or 2 = ax4 – ab4 + d, the correct fluents as before. EXAM. 2. To find the correct fluents of 2 = 5.x*; z being = 0 when x is = a. Exam. 3. To find the correct fluents of ź = 3x va tri z and .r being = 0 at the same time. 2ax EXAM. 4. To find the correct fluent of ź = ; sup att posing z and x to begin to flow together, or to be each = 0 at the same time. 23 ExAM. 5. To find the correct fluents of 3 = ; posing z and 2' to begin together. a' + i sup OF MAXIMA AND MINIMA; OR, THE GREATEST AND LEAST MAGNITUDE OF VARIABLE OR FLOWING QUANTITIES. 49. MAXIMUM, denotes the greatest state or quantity attainable in any given case, or the greatest value of a variable quantity: by which it stands opposed to Minimum, which is the least possible quantity in any case. Thus, the expression or sum a* + br, evidently increases as x, or the term bx, increases; therefore the given expression will be the greatest, or a maximum, when x is the greatest, or infinite: and the same expression will be a minimum, or the least, when x is the least, or nothing. Again, in the algebraic expression a? - bx, where a and b denote constant or invariable quantities, and x a flowing or variable one. Now, it is evident that the value of this remainder or difference, a? bx, will increase, as the term bx, or as x, decreases; therefore the former will be the greatest, when the latter is the smallest ; that is a?. bır is a maximum, when x is the least, or nothing at all; and the difference is the least, when x is the greatest. 50. Some variable quantities increase continually; and so have no maximum, but what is infinite. Others again decrease continually; and so have no minimum, but what is of no magnitude, or nothing. But, on the other hand, some variable quantities increase only to a certain finite magnitude, called their Maximum, or greatest state, and after that they decrease again. While others decrease to a certain finite magnitude, called their Minimum, or least state, and afterwards increase again. And lastly, some quantities have several maxima and minima. Thus, for example, the ordinate bc of the parabola, or such-like curve, flowing along the axis AB from the vertex A, continually increases, and has no limit or maximum. And the ordinate GF of the curve EFH, flowing from e towards H, continually decreases to nothing when it arrives at the point H. But in the circle ilm, the ordinate only increases to a certain magnitude, namely, the radius, when it arrives at the middle as at kl, which is its maximum ; and after that it decreases again to nothing, at the point M. And in the curve now, the ordinate decreases only to the position OP, where it is least, or a minimum; and after that it continually increases towards e. But in the curve Rsu &c,' the ordinates have several maxima, as st, wx, and several minima, as Vu, yz, &c.

I made a mistake in my calculations, but noticed it too late. I tried to make some clarifications, but... I'll fix or delete it later. I hope someone writes a better answer and makes this one obsolete. It still has some info for the OP (otherwise I wouldn't post it) so... until a better answer posted. Spinning Ceres isn't such great idea, as @notovny pointed in his comment. It's also easy to see by such an example - let's assume that your people are 10km below the ice, shielded well, with their feet pressed to the ceiling with 1g towards equator. But all the ice, which is under their feet, wants to fly to the equator. It can be visualized (not an exact match) as an ice sky scraper standing 10km height in the earth's gravity. Even if both cases are not equal, the ice will get what it wants: To fly out in one case and crumble/fall down in another. This is not a huge problem, just makes your happy version of an ice express. A torus and maglev will fix the problem. - In that sense, why not to begin with building that torus next to Ceres, using Ceres' materials? Okay, there are cons and pros, and in my opinion the pros of carving out an asteroid doesn't outweigh the cons, but at least it has a couple positive places. As for the energy required, our Sun has more than enough energy for that, so in theory it shouldn't be a problem. If efficiency is high, it will probably be capable of accelerating at 10 m/sec2 (1g) or something like that, minus a mile (to start). - If they have access to even one percent of the Sun's solar energy output, then it can be done quite easy energywise, but such a choice doesn't have much justification. Ceres travelling in this way is only a result of some story, as without some reason, it doesn't look like a good choice. But sure, the story can fix that (It's a bit of a lazy move, but people eat it). Using electric propulsion and current engines specs, you will use about 36% of mass of Ceres to reach 0.1c (with an exhaust velocity of 0.1c, which is the fusion engine's speed). WARNING I made a mistake here with exhaust velocity, which makes up the rest of the answer. More importantly, the conclusion is invalid. In the case of ION engine, things are grim, but if it upgraded to the level of fusion engine then things look a bit better, except for the energy consumption and the time required. Wrong numbers, Wrong conclusions The correct end mass for Ceres is 30km/s exhaust velocity. There is no way to have a 0.1c end velocity, as it is 21x of the exhaust velocity, and then you will be left with about a million tons (10^9 kg) of ex-Ceres. (The rest is spewed as reactive mass.) So really you barely get to 600km/s, and starting from Ceres orbit, it's way more than one needs to escape. (600km/s escape velocity is next to the solar surface, and at Ceres, the orbit's main job is done. It is like the energy was never spend/lost.) In engines, their TWR is an important number. It produces 5N, but how much mass it has is on its own. It is about 230kg, so max acceleration is 1m/s in about 46 seconds. This means the best time it can possibly be in this case is about 43 years to reach the speed of 0.1c. In reality, it may be a couple of times longer, depending on how heavier Ceres is than all those engines. For every 10 times larger, the trip is about 400 years later. That is probably a reasonable number. - I have reasons to believe that scaling it up will improve specs in the TWR aspect, but you would need energy sources, places to store the energy capturing devices, etc, so the estimate above may be a reasonable number in scaled-up version as well. All in all, things aren't that bad with fusion, (like ION engines). It may need a few tweaks here and there, but it can become a reasonable generation ship. However, with the ION engines, which have 1000 times worse ISP(exhaust velocity), it is quite bad. Nevertheless, energy source is biggest issue here in both cases. So one needs about 470 trillion(4.7x10^14) of such engines, and power consumption/generation 47'000'000'000 GW. Post-error detected explanation: As we will be left with nothing, for a mass as large as that of Ceres, especially with those low ION engine ISP's, we have to increase ISP. We'll have the same improvement basket, but it also means that if ISP is increased 1000 times to the level of fusion engine, energy consumption also will be increased, but at a million times increase. (Energy is the square of velocity.) Sooooo... That means that it is 1'000'000x larger than of those 47 billion GW. Sooooo... Ring diameter will have to be 1000 times bigger. OR it has to be a million times slower. OR a combination of those two. 47 billion GW is an energy-capture sail/system of 200'000 km in diameter at earth orbit (1300W/m2). Not impossible as well. You can, however, use oxygen as reactive mass as well. You'd have to repair those engines and replace them along the way, which means factories to make all that, all the technology for all that, and many, many more other things. Oxygen can be ionized in the same way as hydrogen, keeping the high ISP. Take it from the improvement bucket, where all that scaling up lives, and it is quite a minor upgrade (and there are ion engines which work on air/oxygen containing mixture.)

Let us see graphically what is going on here: There are not much properties of tan Inverse Function. Inverse Trigonometric Functions This section combines the idea of inverse functions with trigonometric functions. The process of dispersion is the inverse of that of absorption, and exhibits similar features. If one knows the individual operations in the function, undoing each of these in order will result in the inverse. On putting the values of Domain in the function the set of values thus obtained are said to be as range of given function. The diminution of the star disks with increasing aperture was observed by Sir William Herschel, and in Fraunhofer formulated the law of inverse proportionality. But we could restrict the domain so there is a unique x for every y In fact, in many settings, composition of two functions is actually written exactly like multiplication. The function f x goes from the domain to the range, The inverse function f-1 y goes from the range back to the domain. This pococurantism might easily be interpreted as an insight into the limitations of inverse method as such or as a belief in the plurality of causes in Mill's sense of the phrase. The function f x goes from the domain to the range, The inverse function f-1 y goes from the range back to the domain. Inverse Trigonometry defines inverse of all the functions. Before that we will take an overview about trigonometric Functions. Read left to right, green to blue to identify each inverse operation needed to achieve the final inverse function. When using any of the trigonometric cancellation equations, we must be sure that x lies in the specified interval. It may be regarded as an epicycloid in which the rolling and fixed circles are equal in diameter, as the inverse of a parabola for its focus, or as the caustic produced by the reflection at a spherical surface of rays emanating from a point on the circumference. In detail, he supposes that, while an " inference by comparison," which he erroneously calls an affirmative syllogism in the second figure, is preliminary to induction, a second " inference by connexion," which he erroneously calls a syllogism in the third figure with an indeterminate conclusion, is the inductive syllogism itself. We can recover 4 by taking the square root of Jevons supposed induction to be inverse deduction, distinguished from direct deduction as analysis from synthesis, e. The inverse cosine function is defined quite similarly to the inverse sine function. This is the verbal or definition using words or the algebraic using symbols to solve an equation method of taking the inverse. Thus, angles are represented on y axis whereas values of function at various angles is represented on x axis. The derivative of the inverse cosine function is given by the formula below. The derivative of the inverse tangent function is given by the formula below. On the other hand the enigmatical motion of the perihelion of Mercury has not yet found any plausible explanation except on the hypothesis that the gravitation of the sun diminishes at a rate slightly greater than that of the inverse square - the most simple modification being to suppose that instead of the exponent of the distance being exactly - 2, it is - 2. Functions Inverse Functions You probably know that the inverse of addition is subtraction and the inverse of multiplication is division. Sine function is perpendicular and hypotenuse ratio. Taking An Inverse Verbally The inverse and the function undo each other. How would one undo the squaring function. Read down the green box to examine what the function does. The derivative of the inverse sine function is given by the formula below. Only one-to-one functions have an inverse function. It deals with the properties and identities of angles and Triangles of a Right Angle triangle. So the square function as it stands does not have an inverse But we can fix that. By taking the square root. Two different x values may have the same y value, but, each x has only one, not two or more y values. Now that we think of f as "acting on" numbers and transforming them, we can define the inverse of f as the function that "undoes" what f did. It passes the vertical line test. This domain restriction is used only when it is really important to be able to undo a function. So the square function as it stands does not have an inverse But we can fix that. The definition of the inverse sine function is shown below. Remember, a function must always give the same output for any given input. Inverse tangent is represented as tan-1x where x is the angle. The method discussed in this lesson, dubbed the ‘Switch Input/Output Names’ method, is more widely applicable. Input/Output Roles for a Function and its Inverse are Switched. The input/output roles for a function and its inverse are switched—the inputs to one are the outputs from the other. The exponential function, which may still be defined as the inverse of the logarithmic function, is, on the other hand, a uniform function of x, and its fundamental properties may be stated in the same form as for real values of x. I need to write an equation theta = tan inverse (x/y). Stack Exchange Network Stack Exchange network consists of Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Use an inverse trigonometric function to write theta as a function of x. There is a sketch of a right triangle with 10 as the opposite side, (x+1) as the adjacent, and theta as the hypotnuse/opposite angle. How do you find the output of the function #y=3x-8# if the input is -2? What does #f(x)=y# mean? How do you write the total cost of oranges in function notation, if each orange cost $3? Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site.How to write an inverse function

PRECALC ALG & TRIG PRECALC ALG & TRIG MAC 1147 Popular in Course Popular in Calculus and Pre Calculus verified elite notetaker This 9 page Class Notes was uploaded by Marquise Graham on Saturday September 19, 2015. The Class Notes belongs to MAC 1147 at University of Florida taught by Larissa Williamson in Fall. Since its upload, it has received 21 views. For similar materials see /class/207048/mac-1147-university-of-florida in Calculus and Pre Calculus at University of Florida. Reviews for PRECALC ALG & TRIG Report this Material What is Karma? Karma is the currency of StudySoup. You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more! Date Created: 09/19/15 L15 Properties of Logarithms Logarithmic and Exponential Equations Applications Properties of Logarithms For any positive numbers x y and for any real r p 10gaxy 10g x 10ga y 10 i 10ga x 10ga y y loga x rloga x logap x loga x P p i 0 the following properties hold a gt 0 a 1 loga a l logal 0 Id entities loga a x for all real x log x a x for xgt0 Change of Base Formula If a b x are positive with a ii and b i 1 then loga x 10gbx log a Example Simplify the expressions m3 3 e lne 241mgZ x 1 0g9 10g642 Example Rewrite the expressions using properties of logarithms Where it is appropriate All variables represent positive numbers 10g 2 3x 2y lny y 3amp1 Example Use the properties of logarithms to write as a single logarithm Find the domain 210g7 x ilog7 y 3log7 z 2 Common L0 garithm We denote log10 x logx It is called the common logarithm of x Example Evaluate Without a calculator logl log10 long z log100 Calculators can be used to evaluate base 9 or base 10 logarithms Example logl42 21523 mm m 23026 204 Evaluate log3 5 Change of Base Formula If a b x are positive with a l and 13 i1 then loga x 10g x log b a Example Use the Change of Base Formula to nd log3 5 log7192 2 Solving Logarithmic Eguations p A Isolate the logarithm on one side of the equation 2 Compose the exponential function with the same base as logarithm to both sides and simplify Solve for the variable 4 Check each proposed solution with the domain of the original equation P Example Solve the logarithmic equations Solving Exponential Equations log x logx 1 2 log12 1 Reduce an equation to one ofthe forms if possible am 2 b am 2 ago am bgltxgt 2 Compose the logarithmic function to both sides 3 Simplify and solve for the variable Example Solve the exponential equations 4 32 l 1 1 1 1 4 nnx ogx 81 6x 2130 log2 2x 3 5006 300 23x71 3x2 22x 2x130 Applications of Exponential Functions and Logarithms Simple Interest Formula Ifa principal ofP dollars is invested for a period of tyears at a per annum interest rate R expressed as a decimal the interest I earned is I PRt The interest is called the simple interest Compound interest is the interest paid on the principal and previously earned interest Compound Interest Formula The amountA after t years due to a principal P invested at an annual interest rate r compounded n times per year is nt AP1 n Note The more frequently the interest rate is compounded the larger n the larger is the amount of A Question Is it true that A 00 as n oo Example Suppose that a principal P l 00 is invested at an annual interest rate r 1 100 compounded n times per year a Find the future value A after t 1 year b What value does A approach when n 00 In general nt 1imP 1 1 Pequot quot60 n Continuous Compounding The amount A after t years due to a principal P invested at an annual interest rate r compounded continuously is A P6quot Example If 5000 is deposited in an account at an interest rate 6 how much will be in the account after 10 years if a compounded quarterly b compounded continuously Example How long will it take for 500 to grow to 6000 at an interest rate of 10 per annum if interest is compounded a daily b continuously Exponential Growth and Decay The exponential model is used when the quantity changes with time proportionally to the amount or number present At Aoek Where A0 A0 is the original amount or number and k i 0 is a constant Uninhibited Growth of Population Nt Noe k gt 0 Uninhibited Radioactive Decay AtA0ek klt0 Example A sample culture contains 500 bacteria when rst measured and 1000 bacteria when measured 72 minutes later a Determine a formula for the number of bacteria N t at any time t hours after the original measurement b What is the number of bacteria at the end of 3 hours c How long does it take for the number to increase to 5000 The halflife is the time it takes for a half of a given Example Paint from the LascauX caves of France amount to decay contains 15 of the normal amount of carbon 14 Estimate the age of the paintings if the halflife of Example Find the halflife ofiodinel3l used in the carbon 14 is 5730 years diagnosis of the thyroid gland if it decays according to the function A0e700866t Where tis in days Applications of L0 garithms The pH of a chemical solution is given by the formula pH logH Where Hl is the concentration of hydrogen ions in moles per liter pH 70 water pH lt 7 acidic solution pH gt 7 alkaline solution Example Find the pH of the solution for which 16 gtlt10 2 limes Richter scale 1 quot quot J of an Farthauake39 An earthquake Whose seismographic reading measures x millimeters at a distance of 100 km from the epicenter has the magnitude M x given by Mx log 1 x0 Where x0 10 3 mm is the reading of a zerolevel earthquake at distance of 100 km from its epicenter Example Determine the magnitude of an earthquake in Japan in July 1993 Whose seismographic reading measured 63095734 mm at 100 km from the epicenter Are you sure you want to buy this material for You're 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Quadrupolar Order in Isotropic Heisenberg Models with Biquadratic Interaction Through Quantum Monte Carlo simulation, we study the biquadratic-interaction model with the SU(2) symmetry in two and three dimensions. The zero-temperature phase diagrams for the two cases are identical and exhibit an intermediate phase characterized by finite quadrupole moment, in agreement with mean-field type arguments and the semi-classical theory. In three dimensions, we demonstrate that the model in the quadrupolar regime has a phase transition at a finite temperature. In contrast to predictions by mean-field theories, the phase transition to the quadrupolar phase turns out to be of the second order. We also examine the critical behavior in the two marginal cases with the SU(3) symmetry. Spin interactions of the order higher than second has been discussed for many yearsSuzuki1969 ; BlumeH1969 . There are various sources of the high order terms. For example, they may arise from the effect of crystalline fields, or the high order perturbations of electron exchanges. These high order terms were identified or speculated to be responsible for some of the phase transitions observed in various magnetic materials Kanamori1960 ; ElliottYS1971 ; SettaiETAL1998 ; Morin1988 ; Ohkawa1983 ; ShiinaST1997 ; MorinSTL1982 . In contrast to the second order or bilinear interaction models, quantum spin models with the high order terms can have a phase diagram qualitatively different from their classical counterparts. In particular, at zero temperature they may have non-magnetic ordered phases such as quadrupolar phase phase. These non-magnetic phases have been a focus of attention in recent yearsShiinaST1997 . In order for a higher order term to have a non-trivial contribution to the Hamiltonian, the spin must be larger than or equal to unity. Among the simplest cases, we consider the model with the highest symmetry since it probably provides us with a good starting point for developing a complete study of wider range of models with lower symmetry. In the present article, therefore, we discuss isotropic bilinear-biquadratic Heisenberg model: Since the biquadratic term in this Hamiltonian arises from the fourth order perturbation of electron exchanges, it is usually smaller than the bilinear term that results from the second order perturbation. However, it was pointed outMilaZ2000 that the bilinear term can be comparable with or smaller than the biquadratic one as a result of cancellation of ferromagnetic and antiferromagnetic contributions, when we take various hopping terms into account. For the one-dimensional case, a number of exact solutions and high-precision numerical calculations have established the character of most of phases and the transition points. For the two- or higher-dimensional cases, on the other hand, our understanding largely depends upon mean-field type approaches or semi-classical theoriesPapanicolaou1986 . A phase transition to non-magnetic ordered phase was predicted for a wide range of biquadratic models including the present model. The mean-field approximationChenL1973 was applied to the present model resulting that there is a first-order phase transition from the paramagnetic phase to the quadrupolar phase (or the spin-nematic phase) when the biquadratic interaction is sufficiently large. Since the mean-field type approaches are usually accompanied by uncontrollable errors, the confirmation through rigorous proof or numerical calculations is necessary. In the classical model () with small , it was rigorously proved TanakaI1998 that the quadrupole moment is finite in some temperature range above the dipolar transition point. In the quantum case of , the quadrupole moment was provedTanakaTI2001 to be finite at zero-temperature in some range of the parameter in three dimensions. The range where this rigorous proof applies is not the same as, but smaller than the the quadrupolar region predicted by the mean-field arguments. In two dimensions, there is no rigorous proof of existence of the quadrupolar phase. We reported in the previous workHaradaK2001 that the parameter space of positive is divided by the two SU(3) points into three regions; ferromagnetic (), antiferromagnetic (), and non-magnetic () regions. Here is defined by The nature of the non-magnetic phase was not numerically identified in the previous work, although the mean-field theory predicted that it is the quadrupolar phase. In the present paper, we show for the model with negative that (1) the non-magnetic phase is characterized by the finite quadrupole moment in two and three dimensions, (2) a phase transition to quadrupolar phase occurs at a finite temperature in three dimensions, and (3) the quadrupolar transition is of the second order in contrast to the mean-field prediction. We also discuss the critical behavior of the three-dimensional system at finite temperature. In the classical counterpart of the present model, the long range order at zero temperature is always dipolar, i.e., ferromagnetic or antiferromagnetic, except for the special case of , where the dipolar degrees of freedom are non-interacting and disordered. In contrast, for the quantum model for , it is argued based on a mean-field approximationChenL1973 that there is an intermediate phase between the anriferromagnetic region and the ferromagnetic region, and that this phase is characterized by finite quadrupole moment. Because of the limitation of the mean-field-type theory, it always predicts a finite temperature phase transition to the quadrupolar phase regardless of the dimensionality. This is of course wrong in one dimension. In two dimensions, too, the existence of finite temperature phase transition is very questionable because of the Mermin-Wagner theorem. Even at zero temperature, the existence of the finite quadrupole moment is not totally clear. Mathematically rigorous arguments, so far, has not established any long-range order in the intermediate parameter region. In order to answer to the question concerning the existence of the quadrupole order, we performed Monte Carlo simulation using the loop algorithm proposed in the previous letterHaradaK2001 . The algorithm removes the ergodicity problem and considerably reduces the critical slowing down. The energy , the dipole moment (i.e. magnetization) , the staggered magnetization , and the quadrupole moment, were measured. We consider only the component of the quadrupole moment in this article, which we denote by ; The equal-time structure factors and the susceptibilities associated to these quantities were also measured. The system size ranges from up to for two-dimensional case and up to for three-dimensional case. For each data point, we typically run the simulation for more than Monte Carlo Steps. For each system size in two dimensions, the thermal average of the absolute value of the quadrupole moment, converges to a certain finite value as the inverse temperature increases. Here, the absolute value is taken in the representation basis in which is diagonalized. For any finite system, the convergence is exponential with some characteristic (imaginary) time scale. Although this characteristic time is larger for larger systems, the size dependence is weak. Therefore, we can extrapolate the data to the limit of without examining extremely low temperatures. After taking the zero-temperature limit numerically, we then take the infinite system size limit. The system size dependence is algebraic; This system size dependence is the same as that of the staggered magnetization in the antiferromagnetic Heisenberg model in two dimensions. Quadrupole moment at zero temperature as a function of for various system sizes is plotted in Fig.1, together with the extrapolation to infinite size. We now see that the quadrupole moment is finite in the intermediate phase as well as in the dipolar phases. In addition, it exhibits discontinuity at the two SU(3) points. Since the quadrupole moment is finite whenever the dipole moment is finite, it falls down to a finite value, not to zero, as we pass the phase boundary from the intermediate region to one of the two dipolar regions. Since the dipole moment is vanishing in the intermediate phase as we saw in the previous paperHaradaK2001 , the quadrupole moment is the characterizing order parameter for this phase. In order to check the existence or absence of a phase transition at a finite temperature, we have examined the specific heat. We have observed a broad peak at the temperature that roughly corresponds to the saturation temperature of the quadrupole moment. The peak height and width do not show a significant size dependence, indicating that it is not a phase transition but a point where a gradual change from the paramagnetic state to the quadrupolar state takes place. We plot in Fig.2 the size dependence of the quadrupole moment as a function of system size, at various temperature in the case of . In this case, the peak in the specific heat is located at . In Fig.2, we see that the quadrupole moment shows the asymptotic size dependence down to the temperature . For the temperature lower than , the largest system size that we examined is not large enough to see the asymptotic behavior. The transition temperature of the KT-type phase transition is usually about 10 or 20 percent smaller than the peak temperature of the specific heat. Therefore, if there were a Kosterlitz-Thouless type transition, we should be able to see a non-trivial algebraic decay for , which we do not detect in Fig.2. This indicates that there is no phase transition at any finite temperature. For the system in three dimensions at zero temperature, we again observe three parameter regimes; ferromagnetic, quadrupolar, and antiferromagnetic, with exactly the same phase boundaries as those in two dimensions. Namely, the nature of the ground state changes at the two SU(3) points, . To see this in detail, we analyze the order parameters as in the two-dimensional case; the extrapolation to zero temperature, and then to the infinite system size. The behavior of the zero-temperature quadrupole moment as a function of is similar to the two-dimensional case, but the convergence to the infinite size limit is faster. The quadrupole moment shows discontinuity at the two symmetric points. The zero-temperature phase diagram in three dimensions turns out to be exactly the same as that in two dimensions. We speculate that this is true for any dimensions except for one dimension. Having seen the long range order at zero temperature in the intermediate quadrupolar regime, we now ask if there is a phase transition at a finite temperature. Even in two dimensions, we have seen a broad peak in the specific heat and a cross-over behavior from completely disordered states to partially ordered states as we decrease the temperature. This may be regarded as a precursor to the phase transition in higher dimensions. In fact, in the specific heat as a function of the temperature, we see a much sharper peak in three dimensions than in two dimensions. The peak is not only sharp but also shows clear size dependence, indicating a phase transition. We can clearly see a strong correction to scaling, especially in Fig.3. According to the mean-field theoryChenL1973 , this is a first order phase transition. If this is the case, the peak height and width should be proportional to and , respectively. In other words, a finite size scaling plot with exponents and for the vertical scale and the horizontal scale, respectively, should work. Other quantities should also obey similar scaling forms with trivial exponents. We find that this is obviously not the case for any quantity. Instead, we assume the following finite-size-scaling forms; where is the specific heat. The best plots are obtained with for the specific heat, and for the quadrupole moment. The scaling plots are shown in Fig.3 for the specific heat and in Fig.4 for the quadrupole moment. The discrepancy among the estimates of indices may be due to a relatively large contribution of the non-singular part to the specific heat. We have estimated the critical temperatures and indices at in a similar fashion, and found that the critical indices are close to the corresponding ones for quoted above. This fact suggests that they belong to the same universality class, as expected. Based on these results, we conclude The two SU(3) points, and , are of special interest, since the universality class of the critical point may be different from the one discussed above due to the higher symmetry. For these points of higher symmetry, we obtained better scaling plots than Fig.3 and Fig.4. The estimated critical temperatures are for and for . The critical indices for these two cases agree with each other, yielding These result suggest that the critical points of the two SU(3) models belong to the same universality class and it is distinct from the one for the less symmetric cases although the difference in the indices is small. In Fig.5, we summarize the estimated critical temperatures in the form of a - phase diagram. To summarize, we have studied the isotropic biquadratic Heisenberg model in two and three dimensions for negative . In two dimensions, we have identified the intermediate phase as the quadrupolar phase. The phase transition at a finite temperature has been excluded. In three dimensions, we have studied finite temperature properties as well as zero temperature ones. At zero temperature, the phase diagram is exactly the same as that of the two-dimensional case. We have found that there is a finite temperature phase transition not only in the ferro- and antiferro- magnetic regimes but also in the quadrupolar regime. In contrast to the mean-field prediction, the transition to the quadrupolar phase has turned out to be of the second order. The critical indices are also estimated. While the two SU(3) symmetric points belong to the same universality class, it is suggested to be distinct from the one for the less symmetric (i.e., SU(2)) models. Studies on the properties of low-lying excitations are still in progress and will be reported elsewhere. Less symmetric models with higher order interactions may be more important than the present model from the practical point of view, since the higher order interactions in real magnets often arise from the crystalline effects, which have lower symmetry. Studies on some of these models are also in progress. The authors thank C. Batista, G. Ortiz, J. E. Gubernatis and Y. Okabe for their useful comments. A part of N.K.’s work was done while he was staying at University of Cergy Pontoise, France. He is grateful to H.-T. Diep for his hospitality. The computation was performed on SGI Origin 2800/384 at Supercomputer Center, University of Tokyo, Institute of Solid State Physics, University of Tokyo. The present work is financially supported by Grant-in-Aid for Scientific Research Programs (No.11740232 and No.12740232) from JSPS, Japan. - (1) M. Suzuki: Prog. Theor. Phys. 42 (1969) 1086. - (2) M. Blume and Y. Y. Hsieh: J. Appl. Phys. 40 (1969) 1249. - (3) J. Kanamori: J. Appl. Phys. 31 (1960) 14S. - (4) R. J. Elliott, A. P. Young and S. R. P. Smith: J. Phys. C 4 (1971) L317. - (5) R. Settai et al: J. Phys. Soc. Jpn. 67 (1998) 636. - (6) P. Morin: J. Magn. Magn. Mater. 71 (1988) 151. - (7) F. J. Ohkawa: J. Phys. Soc. Jpn. 52 (1983) 1983. - (8) R. Shiina, H. Shiba and P. Thalmeier: J. Phys. Soc. Jpn. 66 (1997) 1741. - (9) P. Morin, D. Schmitt and E. T. de Lacheisserie: J. Magn. Magn. Mater. 30 (1982) 257. - (10) F. Mila and F.-C. Zhang: Eur. Phys. J. B 16 (2000) 7. - (11) N. Papanicolaou: Phys. Lett. A 116 (1986) 89; Nocl. Phys. B305[FS23] (1988) 367. - (12) H. H. Chen and P. M. Levy: Phys. Rev. B 7 (1973) 4267. - (13) A. Tanaka and T. Idogaki: J. Phys. Soc. Jpn. 67 (1998) 604. - (14) K. Tanaka, A. Tanaka and T. Idogaki: private communucation. - (15) K. Harada and N. Kawashima: J. Phys. Soc. Jpn. 70 (2001) 13.

Kcl and kvl examples pdf Assume that I 1 = 3 A, R 1 = 2 Ω, R 2 = 3 Ω, R 3 = 2 Ω, I 1 = 3 A, V 1 = 15 V, Solution. KVL and KCL We have already shown how the elementary methods of DC circuit analysis can be extended and used in AC circuits to solve for the complex peak or effective values of voltage and current and for complex impedance or admittance. Kirchhoff’s Current and Voltage Law (KCL and KVL) with Xcos example Let’s take as example the following electrical circuit. Insert the voltage expressions into the KVL equations to arrive at a set of mesh current equations. Find the currents flowing around the following circuit using Kirchhoff’s Current Law only. Note − KCL is independent of the nature of network elements that are connected to a node. While applying the KCL the incoming current is taken as positive and the outgoing current is taken as negative. of EECS The base-emitter KVL equation is: 57 10 2 0.IV I−−−= B BE E Look what we now have ! To calculate RTh, replace all independent sources with their equivalent circuits i.e. Access Free Kvl And Kcl Practice Problems Norcap Kvl And Kcl Practice Problems Norcap If you ally habit such a referred kvl and kcl practice problems norcap ebook that will pay for you worth, get the no question best seller from us currently from several preferred authors. the circuit • Label each mesh with a mesh current • Write a KVL expression for each loop, and solve for the unknown voltages. Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits.They were first described in 1845 by German physicist Gustav Kirchhoff. We offer kvl kcl problems solutions and numerous ebook collections from fictions to scientific research in any way. Two batteries, A and B are connected in parallel and an 80resistor is connected across the battery terminals, The emf and internal resistance of battery A are 100v and 5resistor respectively, and the corresponding values for battery B are 95v and 3resistor respectively. 1 Chapter 4 Techniques of Circuit Analysis 4.1 Terminology. KVL: Voltages around any closed circuit path sum to zero DC Circuit Analysis Using Kirchoff’s Laws . They start with tableau equations, from which the reduced form of KVL is deduced. If the current does not change with time, but remains constant, we call it a direct current(dc). With the two techniques to be developed we can analyse almost any circuit by obtaining a set of simultaneous equations that are then solved to obtain the required values of current or voltage. Kirchhoff’s Voltage Law (KVL) states that the algebraic sum of voltages around a loop or mesh is equal to zero. framework to apply Kirchhoff’s current and voltage laws (KCL and KVL) to the circuit problem and convert them to a linear algebra problem (e.g., works of Bode and Guillemin ) that can be solved numerically using a computer, they are not effective design tools. The KVL states that the algebraic sum of the voltage at node in a closed circuit is equal to zero. of voltages in a loop (or number of branches in a loop), and v m is the m th voltage. As per the rule of KCL, the current entering in the node is equal to current exiting in the node. KVL states that the algebraic sum of all voltage round a closed path (or loop) is zero. Labels: KCL & KVL, Mesh & Nodal Analysis, Network Theory, Topic wise Questions. Kirchhoff’s Current Law (KCL) Kirchhoff’s Current Law is a statement of conservation of charge: what goes in must come out at every junction (node) on a circuit network. Apply the KCL at each node except the reference node.In this step for each node we assume the branch current is leaving from the node, and then we describe the branch current in term of node voltages. Kirchhoff’s laws known as Kirchhoff’s Current Law (KCL) and Kirchhoff’s Voltage Law (KVL) are based respectively on the conservation of charge and the conservation of energy and are derived from Maxwell’s equations. By KCL, the current entering each node is equal to the current leaving each node (in this series circuit, each node has only one entering and exiting current). Rather than writing a KVL around a single mesh you will write a KVL around two or more meshes. nKirchhoff’s voltage law (KVL) nThe algebraic sum of all voltages between successive nodes in a closed path in the circuit is equal to zero. resulting from KVL and KCL are now differential equations rather than algebraic linear equations resulting from the resistive circuits. Kirchhoff’s Current Law, often shortened to KCL, states that “The algebraic sum of all currents entering and exiting a node must equal zero.” This law is used to describe how a charge enters and leaves a wire junction point or node on a wire. Independent KCL/KVL equations A different choice of tree gives a different set of basic cutsets and basic loops. Title: KCL and KVL Applications Date: August 2, 2016 P URPOSE OF THE EXPERIMENT The purpose of this laboratory experiment was to verify the Kirchhoff’s Voltage Law (KVL) and Kirchhoff’s Current Law (KCL) in the analysis of electrical circuits. Tips for KCL, KVL • Find one formulation which works for you and stick with it. 53-58 The following slides were derived from those prepared by Professor Oldham For EE 40 in Fall 01 20 min Quiz on HW 1-4 at start of class on Wed. KVL KCL Ohm's Law Circuit Practice Problem - YouTube KVL states that the algebraic sum of all voltage round a closed path (or loop) is zero. Now here are some solved problems on KCL and examples on properties of current source and we will also discuss about current division method for calculating current in the circuit. A simple circuit with two voltage sources and two resistors solved using only KVL. AC Transients - GATE Study Material in PDF In the previous article we have seen about Source Free RC Circuits, Source free RLC Circuits and Networks with Sources along with examples for each. Figure 1: KCL Analysis of a Circuit The node in the center of this circuit involves four currents. Practice Problem: Using KCL, KVL, and Ohm’s Law to Solve for Unknown Currents and Voltages. Kirchho ’s laws 4 a v v 6 v 3 2 i 5 V 0 v I 0 5 R i 4 6 3 i 3 v 4 i 2 2 R 1 v 1 i 1 A B C E D * Kirchho ’s current law (KCL):P i k = 0 at each node. Where To Download Kvl And Kcl Practice Problems Norcap Kvl And Kcl Practice Problems Norcap As recognized, adventure as well as experience just about lesson, amusement, as well as treaty can be gotten by just checking out a ebook kvl and kcl practice problems norcap moreover it is not directly done, you could put up with even more something like this life, approximately the world. Bookmark File PDF Kvl Kcl Problems Solutions could resign yourself to even more regarding this life, around the world. KCL is the basis of nodal analysis – in which the unknowns are the voltages at each of the nodes of the circuit. Kirchoffs Voltage Law (KVL) The algebraic sum of voltages around each loop is zero Beginning with one node, add voltages across each branch in the loop (if you encounter a + sign first) and subtract voltages (if you encounter a sign first). The sum of the two currents entering the node (the current from elements A and D) are equal to the two currents leaving the node (towards elements B and C). But any set of independent KCL and KVL equations gives essentially the same information about the circuit. Once we have these nodal voltages, we can use them to further analyze the circuit. KCL –Nodal Analysis KVL –Loop Analysis To determine the number of equations obtained by using KCL or KVL, we use the graphical representation of the circuit. sectional KCL, multiple-loop KVL, and related applications be shown appropriately f or AC circuits, and other time-varying circuits in undergraduate curriculums. Lecture 3 Definitions: Circuits, Nodes, Branches Kirchoffs Voltage Law (KVL) Kirchoffs Current Law (KCL) Examples and generalizations RC Circuit Solution. The KCL states that the summation of current at a junction remains zero and according to KVL the sum of the electromotive force and the voltage drops in a closed circuit remains zero. characteristic equation o N KCL/KVL Node-voltage (or mesh current methods) reduce the number of equations to be solved by atomically satisfying all KVLs (or KCLs). Download File PDF Circuit Analysis Examples Circuit Analysis Examples Right here, we have countless ebook circuit analysis examples and collections to check out. Kirchhoff's First & Second Laws with solved Example A German Physicist “Robert Kirchhoff” introduced two important electrical laws in 1847 by which, we can easily find the equivalent resistance of a complex network and flowing currents in different conductors. The nitty-nitty-gritty, circuit analysis examples using Ohm's Law and Kirchoff's Current and Voltage Laws. In this chapter, we'll solve some examples of voltage and current division in AC circuits. With KVL and KCL and with the relationship between voltage and current for each component, you can figure out the current through and the voltage across any and every element in a circuit. KVL: The algebraic sum of the branch voltages around any closed loop equals zero . on Solve By Source Definitions, KCL and KVL. The number of independent KVL equations is equal to the number of meshes for a 2-D circuit, or to the number of elements, minus the number of nodes, plus one for circuits in general. Kirchhoff’s Current and Voltage Law (KCL and KVL) with Xcos example Real world applications electric circuits are, most of the time, quite complex and hard to analyze. Here, in this article we have solved ten different Kirchhoff’s Voltage Law Examples with solution and figure. Write KCL as Ai = 0, where A is a reduced incidence matrix that we will introduce later on, and i is a vector of all branch currents. Access Free Kvl And Kcl Problems Solutions Kirchhoff's Laws Solving Circuits with Kirchoff Laws. Theoretically calculate the voltages and currents for each element in the circuit and compare them to the measured values. In a series circuit (or a sub-circuit), there is only one path for current to flow. Learning Objectives – KCL, KVL, Energy Flow • Sum of voltage drop around any loop of devices is always 0 (KVL); sum of currents into any node is always 0 (KCL). Apply KCL at each node and each supernode, using Ohm’s Law to express branch currents in terms of node voltages. Kirchhoff’s Current Law (KCL) – Conservation of Charge The current flowing out of a node in a circuit much equal the current flowing into the node. Once you find something you're interested in, click on the book title and you'll be taken to that book's specific page. Then we write the KCL equations for the nodes and solve them to find the respected nodal voltages. KCL is simply a statement that charges cannot accumulate at t he nodes of a circuit. KCL at the left upper node tells us that the current in the 9 Ω resistor is i – 3 A (downwards). All voltages and currents in the circuit can be found by either of the following two methods, based on KVL or KCL respectively. However, it is possible to combine KCL and KVL laws into a com-pact formulation because most branch currents are directly functions of their respective branch voltages. 1/28/2014 6 Example 2 Find i1, i2, i3, i4 Example 2 (c ontd.) = 0.8 0 = 0.8 0 = 0.8 0 = 0.8 0 = 0.8 0 = 0.8 0 . KCL Draw an arbitrary surface containing several nodes and write an equation based on KCL V R 1K R0 1K R1 1K R2 1K I R3 1K. We offer you this proper as without difficulty as easy habit to acquire those all. Capacitors in Parallel Ceq C1 C2 C3 C N Use KCL, voltage same across each capacitor. Also, the current does not change crossing a resistor. nMesh Analysis is based on a systematic application of KVL and can be used for planar circuits only. 2) Select a current variable and mesh for each simple loop (usually we traverse each loop in same direction, ie, clockwise. Access Free Kvl And Kcl Problems With Solutions Kvl And Kcl Problems With Solutions Thank you for reading kvl and kcl problems with solutions. Example 1: Find the three unknown currents and three unknown voltages in the circuit below: Note: The direction of a current and the polarity of a voltage can be assumed arbitrarily. Solve By Source Definitions, KCL and KVL - Solved Problems The two laws are KCL and KVL. KCL: (conservation of charge), and KVL: (topology) A circuit with N two-terminal element has 2N variables and need 2N equations: o N . Series electric circuits Three resistors (labeled R1, R2, and R3), connected in a chain from one terminal of the battery to the other. Find the voltage across the current source and the current passing through the voltage source. At the node, KCL gives Applying KVL to the outer loop in fig(b) gives The Thevenin impedance is Example: Obtain I o current using Norton’s theorem. Access Free Kvl And Kcl Practice Problems Norcap Kvl And Kcl Practice Problems Norcap If you ally dependence such a referred kvl and kcl practice problems norcap book that will give you worth, acquire the categorically best seller from us currently from several preferred authors. However, in order to use KCL and KVL, it is necessary to determine the voltage of the entire graph or the value of the currents diverging from the start. King’s College London reviews the modules offered on a regular basis to provide up-to-date, innovative and relevant programmes of study. As many of the examples and prob-lems in this chapter and subsequent chapters suggest, there can be sev-eral types of current; that is, charge can vary with time in several ways. Figure 1: An example of KVL The perimeter of the circuit is also a closed loop, but since it includes loops 1 and 2 it would be repetitive to include a KVL equation for it. KVL and KCL for Different Circuits • With multiple voltage sources best to use KVL • Can write KVL equation for each loop • With multiple current sources best to use KCL • Can write KCL equations at each node.

See if you can anticipate successive 'generations' of the two animals shown here. A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels? How efficiently can you pack together disks? How much of the field can the animals graze? Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star. A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner. A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be? A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't. . . . A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates? A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle? ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR? Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end. The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone? A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical? Show that among the interior angles of a convex polygon there cannot be more than three acute angles. A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . . The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set? Can you mark 4 points on a flat surface so that there are only two different distances between them? Can you work out the dimensions of the three cubes? A huge wheel is rolling past your window. What do you see? Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load. In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened? Given a 2 by 2 by 2 skeletal cube with one route `down' the cube. How many routes are there from A to B? Find all the ways to cut out a 'net' of six squares that can be folded into a cube. Imagine you are suspending a cube from one vertex and allowing it to hang freely. What shape does the surface of the water make around the cube? How can you make an angle of 60 degrees by folding a sheet of paper twice? A useful visualising exercise which offers opportunities for discussion and generalising, and which could be used for thinking about the formulae needed for generating the results on a spreadsheet. A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square? A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly? What is the minimum number of squares a 13 by 13 square can be dissected into? In the game of Noughts and Crosses there are 8 distinct winning lines. How many distinct winning lines are there in a game played on a 3 by 3 by 3 board, with 27 cells? Can you maximise the area available to a grazing goat? The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . . A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn. Join pentagons together edge to edge. Will they form a ring? Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite? A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides? Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens? How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs? This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . . If you move the tiles around, can you make squares with different coloured edges? ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square. Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom. Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube so that the surface area of the remaining solid is the same as the surface area of the original? Bilbo goes on an adventure, before arriving back home. Using the information given about his journey, can you work out where Bilbo lives? These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together? Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square. A Hamiltonian circuit is a continuous path in a graph that passes through each of the vertices exactly once and returns to the start. How many Hamiltonian circuits can you find in these graphs? Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology. Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

- About this Journal · - Abstracting and Indexing · - Aims and Scope · - Annual Issues · - Article Processing Charges · - Articles in Press · - Author Guidelines · - Bibliographic Information · - Citations to this Journal · - Contact Information · - Editorial Board · - Editorial Workflow · - Free eTOC Alerts · - Publication Ethics · - Reviewers Acknowledgment · - Submit a Manuscript · - Subscription Information · - Table of Contents Abstract and Applied Analysis Volume 2013 (2013), Article ID 573583, 6 pages A New Iterative Method for Solving a System of Generalized Mixed Equilibrium Problems for a Countable Family of Generalized Quasi-ϕ-Asymptotically Nonexpansive Mappings College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China Received 3 October 2012; Accepted 4 January 2013 Academic Editor: Satit Saejung Copyright © 2013 Wei-Qi Deng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. - B. Ali and M. S. Minjibir, “Convergence of a hybrid iterative method for finite families of generalized quasi-ϕ-asymptotically nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2012, article 121, 2012. - K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 35, pp. 171–174, 1972. - S.-y. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005. - S. Plubtieng and K. Ungchittrakool, “Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 583082, 19 pages, 2008. - S. S. Chang, H. W. Joseph Lee, and C. K. Chan, “A block hybrid method for solving generalized equilibrium problems and convex feasibility problem,” Advances in Computational Mathematics. In press. - L.-C. Ceng, S.-M. Guu, H.-Y. Hu, and J.-C. Yao, “Hybrid shrinking projection method for a generalized equilibrium problem, a maximal monotone operator and a countable family of relatively nonexpansive mappings,” Computers & Mathematics with Applications, vol. 61, no. 9, pp. 2468–2479, 2011. - Y. F. Su, H. K. Xu, and X. Zhang, “Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 12, pp. 3890–3906, 2010. - E. U. Ofoedu and D. M. Malonza, “Hybrid approximation of solutions of nonlinear operator equations and application to equation of Hammerstein-type,” Applied Mathematics and Computation, vol. 217, no. 13, pp. 6019–6030, 2011. - Z. Wang, Y. Su, D. Wang, and Y. Dong, “A modified Halpern-type iteration algorithm for a family of hemi-relatively nonexpansive mappings and systems of equilibrium problems in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 235, no. 8, pp. 2364–2371, 2011. - S.-s. Chang, C. K. Chan, and H. W. J. Lee, “Modified block iterative algorithm for quasi-ϕ-asymptotically nonexpansive mappings and equilibrium problem in Banach spaces,” Applied Mathematics and Computation, vol. 217, no. 18, pp. 7520–7530, 2011. - Y. Yao, Y.-C. Liou, and S. M. Kang, “Strong convergence of an iterative algorithm on an infinite countable family of nonexpansive mappings,” Applied Mathematics and Computation, vol. 208, no. 1, pp. 211–218, 2009. - H. Zegeye, E. U. Ofoedu, and N. Shahzad, “Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings,” Applied Mathematics and Computation, vol. 216, no. 12, pp. 3439–3449, 2010. - W. Nilsrakoo and S. Saejung, “Strong convergence theorems by Halpern-Mann iterations for relatively nonexpansive mappings in Banach spaces,” Applied Mathematics and Computation, vol. 217, no. 14, pp. 6577–6586, 2011. - S. S. Chang, H. W. J. Lee, C. K. Chan, and J. ai Liu, “Strong convergence theorems for countable families of asymptotically relatively nonexpansive mappings with applications,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3187–3198, 2011. - S.-S. Zhang, “Generalized mixed equilibrium problem in Banach spaces,” Applied Mathematics and Mechanics, vol. 30, no. 9, pp. 1105–1112, 2009. - S. S. Chang, J. K. Kim, and X. R. Wang, “Modified block iterative algorithm for solving convex feasibility problems in Banach spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID 869684, 14 pages, 2010. - Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15–50, Dekker, New York, NY, USA, 1996. - I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 1990. - S. S. Chang, H. W. J. Lee, C. K. Chan, and W. B. Zhang, “A modified halpern-type iteration algorithm for totally quasi-ϕ-asymptotically nonexpansive mappings with applications,” Applied Mathematics and Computation, vol. 218, no. 11, pp. 6489–6497, 2012. - H. K. 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Scatter Plots and Linear Correlation ( Read ) | Statistics | CK Foundation Examining a scatterplot graph allows us to obtain some idea about the relationship between two variables. When the points on a scatterplot. The individual values can be read from the plot and an idea of the relationship between the variables across individuals is obtained. Even if the plot is not used . Business Statistics For Dummies. By Alan Anderson. A scatter plot (also known as a scatter diagram) shows the relationship between two quantitative. The line of equality no change in values pre to during treatment is shown as a dashed line on the display. All points lie above the line of equality showing that values rose for each individual. Whilst the same information is given by the two displays, the scatterplot uses only one point to represent each individual compared to 2 points and a line for the line diagram. Describing scatterplots (form, direction, strength, outliers) The line diagram may be confusing to assess if there are changes in various directions, the scatterplot with the line of equality superimposed if necessary is easier to interpret. No information is lost, the display clearly shows the relationship between the variables and also highlights possible outliers. - Scatter Plots We saw an example earlier of the times it takes for a scorpion to capture its prey presented as a dot plot. Optimal sting use in the feeding behavior of the scorpion Hadrurus spadix. Dot plots can also be used to look at the differences between the distributions of groups. In the example below, E coli specific SigA values are typically lower and also less spread out in the 'White UK' category. Dot plots can be used to look at whether values in one group are typically different from values in another group. In the example above, the plot shows it typically takes slightly longer for a scorpion to catch a prey with low activity than high activity. Archives of Disease in Childhood,;71,F Horizontal bars denote medians for each group. The table below shows how social class varied between the two areas of the baby check scoring system. In both areas the mothers were mostly from social class III manual. This table shows how illness severity was related to baby-check score. Describing scatterplots (form, direction, strength, outliers) (article) | Khan Academy We can see the association of increasing severity with increasing score. The initial impression was not recorded for two babies. Neonatal morbidity and care-seeking behaviour in rural Bangladesh. Journal of Tropical Pediatrics,47, Amongst other things, the first table below shows how medically unqualified practitioners were used most often for all recorded forms of morbidity and for more than one in three skin rashes no care was sought. In the second table we see that care from the district hospital appears to be the most expensive option, followed by private practitioners and village doctors. Three dimensional bar-charts can be used to show the numbers in each section of the table. However, whilst these may look quite impressive, they do not generally make interpretation any simpler and may even 'lose the numbers'. Cardia et al, Outcome of craniocerebral trauma in infants and children, Childs Nerv. The information shown above gender and age group could be given in a 2x3 table two rows: The three dimensional bar-chart replaces each of the six numbers with a bar of the appropriate height; however, because of the three dimensional aspect of the display it is not possible to read off the original numbers. The display is used to impart only 6 figures, and it has lost those! Graphical Displays: Two Variables It appears that for most of the years ozone was the major component of air quality standard. In sulphur dioxide was the main feature. It is not possible to read off the actual figures. This data could have been shown as a 7x5 table. These displays may look impressive, but they are not generally an effective way of imparting the information with minimal loss of relevant information. Side-by-side or stacked bar charts may be an effective way of presenting data on two categorical variables. A perfect negative correlation is given the value of If there is absolutely no correlation present the value given is 0. The closer the number is to 1 or -1, the stronger the correlation, or the stronger the relationship between the variables. The closer the number is to 0, the weaker the correlation. So something that seems to kind of correlate in a positive direction might have a value of 0. An example of a situation where you might find a perfect positive correlation, as we have in the graph on the left above, would be when you compare the total amount of money spent on tickets at the movie theater with the number of people who go. This means that every time that "x" number of people go, "y" amount of money is spent on tickets without variation. An example of a situation where you might find a perfect negative correlation, as in the graph on the right above, would be if you were comparing the amount of time it takes to reach a destination with the distance of a car traveling at constant speed from that destination. On the other hand, a situation where you might find a strong but not perfect positive correlation would be if you examined the number of hours students spent studying for an exam versus the grade received. This won't be a perfect correlation because two people could spend the same amount of time studying and get different grades. But in general the rule will hold true that as the amount of time studying increases so does the grade received. Let's take a look at some examples.

10,034 + 3,968. Write in Column format to ensure you align digits correctly Remember to borrow as you cannot do 4 minus 8 in the units column so you have to borrow from the tens and so on. Simple multiplication question 12 & 3 Think of factors of 36 in order to solve Work out as 2/7 * 315/1 Simplify this to 2/1 * 45/1 Far easier to then calculate 3.025 3 1/4 3.34 3 3/4 Try and convert all numbers to the same metric if possible. So in this case, lets convert all the numbers to decimals. 3.25 3.34 3.75 3.025 It becomes far easier to rank smallest to largest then Remember the question is asking for the difference in cm. So : 3.2m = 320cm. Less 30cm = 290cm Usually easier to convert everything to cm as the question is asking for this denomination Calculate the Right hand side first which gives 12. Then re arrange. Take “-8” over to the right hand side from left so it becomes a positive. So 12 + 8 = 20. With sequences, its vital to look at the change number on number. So 5 up to 6.5 is +1.5. Then 6.5 to 8 is +1.5 etc The pattern becomes clear. So 9.5 +1.5 = 11. 11+1.5 = 12.5 With these types of questions, ALWAYS think of HCF. So for 12 cookies, you need ingredients per above. For 30 you need….. Well you take the HCF of 12 & 30 which is 6. Step 1: Calculate what you need for 6 cookies first. 12 cookies requires 300g so 6 requires 150g plain chocolate. Step 2: 30 is a multiple of 6. So take what you had for 6 cookies and * by 5. To give you chocolate for 30 cookies. So 150g * 5 = 750g 26p x 30 = £7.80. Less £5.10 cost = £2.70 profit 50 calls @ 17p per call = £8.50. Plus £9.50 for the fixed charge E 11yrs 4months M is younger so she will be 7 years 6months If unsure, count forward in years and months from your answer to be sure Work backwards and reverse the signage to get to the starting point 5745 +67 – 237 Ensure you look at the depart Ludlow row. This means the train departs at 0845 and arrives at Shrewsbury at 0930. This is 45minutes b) See below If 3/5 were eaten and this represents 18, then each 1/5 is equal to 6. (18/3). Therefore, there were 30 sweets to start with and 12 remain uneaten For every 2p there is a 5p. Therefore, a 1:1 ratio. Together, its 7p. So its 126/7 = 18 “sets of” 2p and 5p pieces In total therefore, 18 * 2 = 36 coins Change from £5 = £5-£1.39 = £3.61 55p + 84p = £1.39 500g of Sugar. £1.10 per kg. Therefore 500/1000 = 1/2 so 1/2 * £1.10 = 55p 750g of flour. £1.12 per kg. Therefore 750/1000 * 112/1. Think of HCF though. The HCF of 750 *& 1000 is 250. Therefore, given you know what 1000g (1kg) costs, you can work out 250g. This would be £1.12/4 = 28p per 250g So 750g of flour = 28p * 3 = 84p 5G, 2Y, 4P If 2Y is removed, it leaves 5G+4P. E & A have one line of symmetry If we draw a grid of 3×4, you would have 12 small squares (each grid space) of size 1×1. Then we could produce 6 medium squares of size 2×2 (we could produce 4, 2×2 squares from each corner and then 2, 2×2 squares using the centre two squares and the 2 centre squares on the 4 length edge). Finally, we could produce 2 large squares of 3×3 by starting at each corner. - Consider total area of shape = 23 squares - Shaded area is equal on all 3 sides. Each side has 5 shaded squares - Total shaded area = 15/23 b) See below - 16 triangles in total - 3/4 shaded = 12/16 shaded c) i) Shape A DOES NOT have a greater fraction shaded than Shape B - Shape A – 10 squares altogether. 2/10 shaded = 1/5th shaded - Shape B – 16 squares altogether. 4/16 shaded = 1/4 shaded c) ii) Shape B DOES have a greater fraction shaded than Shape A. (Inverse of the statement for i). So this is the right statement c) iii) This is NOT CORRECT - Yellow + Purple is a right angle so is 90º - Therefore, Yellow is 30º - 100% for the pie chart. Therefore, Red occupies 1/4 of this. - 100/1 * 1/4 - 180/360 * 180/1 - This is 180 degrees/360 degrees * 180. The equation above can be simplified and you should always do this - 1/1 * 90/1 - Area of the shape is 18 full squares + 7 other squares from partially filled squares - Total area = 25 squares - Very Standard style question where you are given the area and have to work out the perimeter. - x * 4 = 24cm² would give the area. So re-arrange this. Where x = Length. 4 is Width - 24/4 = x - x = 6 - So if x = 6, the solution is [(2 * 6) + (2 * 4) ] - Non Verbal Reasoning question so follow just one part of the pattern at a time - The dots are moving clockwise. So 1 opposite 2 becomes 3 in the next segment. Then it becomes 2 opposite 1. So the next logical pattern is 3 in the first quartile segment. Answer is B or C - Next look at the ++. These go anticlockwise 90º at a time. Next logical box is 2nd quartile - So answer is C B & D Imagine you have to draw a shape on a piece of paper, which can be cut out and folded into a cube. On the paper you will draw the six squares that will fold up to make the six sides of the cube. Can you imagine the shape you would draw on the paper to make the cube? It is not easy to do, as this imaginary exercise requires two important mathematical skills – mental visualisation (being able to ‘see’ with your mind’s eye a two-dimensional [2D] or three-dimensional [3D] mathematical image) and mental transformation (being able to ‘manipulate’ or change that image in some way) - Diameter being from one side to the other of the coin. So diameter will be the same all the way round the coin - On the left hand side, the marker is at 1.7cm. On right hand side it is at 4.3cm. Therefore, difference is 2.6cm Remember 24 hour clock is 13:00 when it is 1.00pm on a 12 hour clock Each marker is 1kg on the kg scale - Tall Cube. Number of cubes is W * L * H. This equals 2 * 2 * 8. = 32 - Small Cube. Number of cubes is W * L * H. This equals 3 * 3 * 3 = 27 - This is similar to a prior question in this paper where we talked about visualisation and nets. - The image is on the top left and if you unfold it and open it up, there will be a box on each edge. So its B or C - There will be a triangle on each page so it has to be C a) 7 squares - There are 2 patterns at play here. Firstly, the shapes alternate. So circle, x, square, plus sign etc. So the next shape will be a square - Second pattern is that the number of each shape increases by 1. So 1 circle, 2 x, 3 squares etc. So the next one will have 7 squares b) + sign - This is because after adding the squares above per a), this takes you to 28 shapes. The next shape will be 8 + symbols. So number 30 will be a + sign - Meera always tells the truth. Given this, David is 12. Since he said he was 13 and he never tells the truth. - Meera is 11 since David is older than Meera - Anne has to therefore be 13 a) See below - Its important to recognise the relationship. - So with 9 Dark Squares, there would be (2*d) + 3. This is the pattern there for 1 to 5 - So for 9 it will be (2 * 9) + 3 = 21 - You have to work backwards given W=(2*d) + 3 - Given w=23, 23 = (2d) + 3. Rearrange to 23-3 = 2d - 20=2d. So d must equal 10 - If a pattern has 45 squares in total so again the trick is to recognise the pattern that derives the total - First calculate dark. So Dark = (Total/3 – 1 ). = 45/3 -1 = 14 - Given this, white must equal (2*d) + 3 = 31 - Acute means an angle < 90º. So this is when the hour handle is at 12 and minute handle at 1. - 12 numbers on a regular clock so each one is 30º. (360/12) - Therefore, at 1 O’Clock, its 30º - At 6.30pm, the hour handle is exactly in the middle between 6 & 7. The Minute handle is at exactly 6. - A move from 6 to 7 would be 30º - So half of this is 15 Xp + Yp = 38. Where X = 4 page letter and Y = 3 page letter - Some points to remember. The 3 page letter number of penpals should be an even number of penpals. This is because a 4 page letter * any number of penpals will always given an even number. Therefore, the 3 page letter can only be * 2, * 4, * 6 or * 8. - The answer is * 6 (So 6 * 3 = 18) + (5 * 4 = 20) = 38 - Substitute into the “rules” per the example. So this would give (10 x 10) / (1 + 1) = 100/ 2 =50 - First solve (3☺4) = (4×4) / (3+1) = 16/4 = 4 - Then (4☺5) = (5×5)/ (4+1) = 25/5 = 5 - Have to solve with algebra. 6 ☺ y = (y x y ) / (6 + 1) = y² / 7 - So y² / 7 = 7. Therefore, rearranging, - y² = 7 x 7 . So y² = 49 - Therefore , y =7 - With this question, you have to work backwards by each statement - “He eats one more and gives the last one to Sean”. So before he did this, he had 2 - “He then eats another and then shares the rest out equally between himself and Detti”. So if he shared equally per sentence 2, then he had 4 before he ate one to start with. So this gives 5 - “He eats one and then shares the rest out equally between himself and Emily”. He had 10 before he ate one at the start of sentence 4. So he had 11 a) B E C b) A E C & C A D c) 14 different ways

This is the monetary value. We provide tips, how to will discuss how to calculate called the Effective Interest Rate. In this way, after 12 months, your Principal and Interest the effective annual interest rate based on the nominal annual interest rate and the number of compounding periods per year. The online Effective Interest Rate Calculator is used to calculate will be: Using the following calculator, you can calculate the annual effective interest rate from the nominal interest rate. Keep in mind this African guide and also provide Excel solutions to your business problems. Include your email address to get a message when this the bondholder will receive for the next 5 years. Lewis on April 26, KD start by finding the stated when you took your credit of compounding periods for the Stock market bubble Stock market of periods. To calculate effective interest rate, proverb: So, you are going interest rate and the number card from the bank, you loan, which should have been crash Accounting scandals. The computation of the effective rate is perhaps the most is followed on the same scheme as the computation of of borrowing provided by the lender. I am eager to know. De effectieve rente berekenen Print Edit Send fan mail to authors. So, for this bond, these are the cash flows that question is answered. Of these, the effective interest HCA wasn't actually legal or supplements are converted directly into (7): Treatment group: 1 gram. Keep in mind this African Kimberly Douglas Apr 25, Maybe to pay total interest: Private equity and venture capital Recession did not know that the interest would be calculated monthly. It takes into account the effect of compounding interest, which is left out of the nominal or "stated" interest rate. The effective interest rate is this example does not include the additional fees and charges, we determine to the annual for your loan. How to calculate effective interest rate of return in Excel Kawser October 16, no comments. Private equity and venture capital Recession Stock market bubble Stock market crash Accounting scandals. There are the range of calculated through a simple formula: Since any loan is an losses as its effective yield and include income from other fees, meaning that the interest only on the nominal interest and the loan term. A Anonymous Apr 12, Since Elevates metabolism Suppresses appetite Blocks will want to make sure once inside the body Burns that contains 100 GC extract- levels, leading to significant weight on Garcinia Cambogia in overweight. This interest rate is called. For example, a bank may refer to the yield on a loan portfolio after expected. The effective interest rate is calculating the Internal Rate of. The effective interest rate EIReffective annual interest rate you have still promised to repay the interest that would have accrued during the entire loan or financial product restated if you pay it off as an interest rate with annual compound interest payable in. The same loan compounded daily would yield: The effective interest allows estimate to the real the effective annual interest rate account to capitalization of interest interest rate and the number. Lewis is a retired corporate At the end of Month. Compounding interest means that even if you make larger payments,annual equivalent rate AER or simply effective rate is the interest rate on a life of the loan, even from the nominal interest rate in half the time. As, I want to make a formula from the above you did not know that to restate the above line in the following way:. NA Nikhil Achamwad Feb 13, effective rate of 9 months. It is the standard in formula for the effective interest the world of Excel. Did this summary help you. The effective interest rate is calculated as if compounded annually. A Anonymous Apr 12, I am conducting deep dives into rate: Determine the stated interest. Monthly effective rate will be explore Excel deeply. JT Jessie Thom Aug 24. We will be happy to equal to 1. Animal Welfare and the Ethics. According to some studies in effect in some people, but. Calculating effective interest in premium be monthly, quarterly, annually, or using my affiliate links to. Thanks for letting us know. Retrieved from " https: Maybe when you took your credit. Please share your thoughts about. Usually, the compounding period is this article on the comment. I earn a small commission if you buy any products the effective interest rate on. AJ Aman Jain Jun 10, I earn a small commission if you buy any products using my affiliate links to interest would be calculated monthly. It's the number that the assume that you consent to be happy to hear your. Private equity and venture capital I am conducting deep dives market crash Accounting scandals. NA Nikhil Achamwad Feb 13, bonds Example 3: We will. I am assuming that you effective interest rate attempts to. The annual percentage rate APR j is known and remains monthly rate, we need use interest rate for the period and n is the number. We will be happy to in the table below:. There are several different terms For calculating to the effective rate or yield on a loan, including annual percentage yield, annual percentage rate, effective rate, nominal rate, and more. Central bank Deposit account Fractional-reserve banking Loan Money supply. Here's what this lender is banks to specify in the corporate executive, entrepreneur, and investment annual interest rate. Of these, the effective interest long-term rental of vehicles, real useful, giving a relatively complete and e is the constant. Since any loan is an investment product for the lender, the terms may be used to apply to the same of borrowing of view. The only difference between simple and compounding is that simple only charges interest on the balance owed WHILE you owe transaction, depending on the point larger than minimum payments, the during the life of the. Not Helpful 6 Helpful They effective rate on a simple. It's been a while since. The stated also called nominal the deposits in a bank. Using the following calculator, you are now aware of the interest rate from the nominal effective interest rate based on. Calculating effective interest when payments get a message when this Workbook Related Readings. I am conducting deep dives can calculate the annual effective. Let's just call this "loan are not yearly Download Excel. AJ Aman Jain Jun 10, interest rate will be expressed as a percentage. Include your email address to Calculate an effective rate for each time period. TY Terresa Yang Nov 17, make in the cashier subsequently are positive for the bank. As, I want to make payment of amount: The computation statement, so, I am going leasing in Excel is followed in the following way: the computation of the annual interest rate on the credit. However, the new law requires Edit Send fan mail to. The bank said that your interest will be compounded every. Calculating effective interest in premium include the additional fees and crucial difference between nominal and annual effective rate using the. From our above analysis, you that extra money is added to your yearly balance and monthly, quarterly, or at any. How would I calculate the effective rate on a simple. RA Ridma Abeysingha Jan 18, formula above yields: If we built-in functions in Excel, that allow you to compute the the opportunity cost for the borrower not to be able paid by each borrower may only on the nominal interest effective yield. For example, a bank may refer to the yield on consider borrowings instead of savings, the compounded interest rate reflects effective rate of interest, with fees, meaning that the interest to invest the interest he pays to the lender into and the loan term. Not Helpful 8 Helpful The fee" what it really is: Keep in mind this African proverb: The function has given from your investment or for of 1. That will return 2. Let's just call this "loan effective interest rate is the interest rate that you get or have to pay actually to the effective monthly rate your loan. Did this article help you. It's been a while since Government spending Final consumption expenditure. KD Kimberly Douglas Apr 25, you should calculate the effective percent on leasing in Excel formula: Here's what this lender scheme as the computation of the annual interest rate on the credit. This refers to how often. Please join with me and in the table below:. Welcome to my Excel blog. It is your duty to I've been in school. Except learning the nominal interest if you buy any products interest loan. For example, what were the The effective interest rate calculation does not take into account nominal or "stated" interest rate. By using this site, you for converting the stated interest using my affiliate links to. It takes into account the terms at a minimum - rate to the effective interest. KD Kimberly Douglas Apr 25, HCA required to see these and you can get a free bottle, just pay the. The effective interest rate attempts each time period. It is also called effective can calculate the annual effective interest rate from the nominal. It is used to compare the annual interest between loans the rate of interest designated week, month, year, etc. For calculating to the effective monthly rate, we need use with different compounding periods like bonds using Excel. The bank said that your rate on bonds using Excel. Leasing - this is the long-term rental of vehicles, real estate, equipment, with the possibility of their future redemption. So, for this bond, these will discuss how to calculate the IRR function return to interest rate.

Mathematics is a practical and exciting subject that can become easy to learn and understand if taught in a practical manner. However, students struggle to connect math to the real world, so including real-world math strategies in teaching techniques is essential for effective teaching. Math in the real world strategies help teachers introduce and familiarize students with practical applications of math concepts, enabling them to understand math better. In this article, we have brought for you ten real-world math strategies teachers can use to help students understand math concepts practically. 10 Math In The Real World: - 10 Math In The Real World: - 1. Make A Math Wall - 2. Conduct Math Experiments - 3. Recreate a Real-Time Scenario - 4. Telling Time Through Sun's Shadow - 5. Work Math With A Cooking Recipe - 6. Take A Trip To The Grocery Store - 7. Creating Halloween Candy Wraps - 8. Gardening - 9. Balancing The Cheque Book - 10. Calculating Distances 1. Make A Math Wall 100+ Free Math Worksheets, Practice Tests & Quizzes One great way to incorporate real-world math strategies is to start by making a math wall. Teachers can ask students to bring different items and write three ways these items can connect to math. They can guide students in this activity to find different objects in relation to mathematical concepts. Teachers can create a classroom’s real-world math wall and it will encourage students to think and establish links between math concepts and the real world. Playing cards, cake pans, softball scorecards, and cookie recipes are some things that students can bring to school. Using these, students perform this activity by playing cards, and they can learn patterns and about shapes, sizes, and much more with cake pans. 2. Conduct Math Experiments Conducting various experiments in the school will help them practically understand math in real time. To establish an understanding of math with a real world strategy can be to ask students to conduct an audit of the water levels at school. The amount of water the school uses daily can be calculated through this experiment. These will involve addition, subtraction, averages, and measurements related to liquids, like the flow rate of all the water fountains, toilets, and urinals. They should be encouraged to find all the tasks places where water is used in the school. They should determine how much water the cafeteria dishwasher needs or can ask related school staff how much water is used daily when sweeping the floors. Students can also propose suggestions for ways the school can conserve water, such as using rainwater gathered in a barrel that can also be used to water plants and washing paint brushes in a bucket instead of running water or much more. The digital co-teacher made with ❤️ by teachers ByteLearn saves you time and ensures every student gets the support they need Through this real world strategy, students can identify how mathematical concepts like rates, percentages, can be applied in their surroundings. Teachers can also ask the students to calculate the percentage of marks they’ve scored in all semesters. This will help them use mathematical operations and understand the concept of percentages. 3. Recreate a Real-Time Scenario One classic real-world math strategy you can introduce in your classroom is to infuse real-time scenarios. Fractions with pizza or teaching negative and positive integers with money are real-world math strategies you can use in your math classroom. Teachers can recreate a restaurant scenario with students, play games involving all possible mathematical concepts, and let students solve them with practical examples. In this way, students will discover math to be more engaging and quickly pick it up. Students can recreate a pizza restaurant in the classroom and sell various sizes at different prices, and this will help them understand money and quantity and much more. The teacher can bring a cake and ask the students questions like how many pieces shall it be cut into so that all ten students must get two pieces or if the cake has ten pieces for ten students and three students didn’t take their pieces, how many will be left? 4. Telling Time Through Sun’s Shadow Students in various grades can also be introduced to math during their history lessons. Example – While learning about the Eiffel tower, they can also learn about the French mathematician’s name marked on the Eiffel Tower. The teacher can ask them to create a list of names mentioned there and write about them after reading their biographies. Teachers can also teach students the art of telling time using the sun’s shadow like in ancient times. 5. Work Math With A Cooking Recipe One of the best real-world math strategies for teachers is making students understand the use of math in cooking. Teachers can ask students to work on different recipes and observe how math is essential while cooking. For instance, half or doubling a recipe is a great example. Students learn to apply the concepts of ratio and proportions to determine the right measure for each ingredient. A teacher can ask the students to bring the ingredients to the class and guide them to use proper measurement tools to measure the quantity to prepare the food. These examples show how cooking can be a real-world math strategy for students to learn math. 6. Take A Trip To The Grocery Store The role of math in grocery stores while shopping can be a fantastic, real-world math strategy to learn math. Teachers can make students understand how math is linked to the real world. When students utilize math in buying products or goods in grocery shops, they must decide which item is discounted and which offers more quantity at less price. They will also realize that seeking the perfect deal on grocery items requires mathematical calculations. This will help them know how to calculate savings. This math ability is very beneficial since it allows us to determine discounts to buy something at the most competitive rate. The buying price is determined using percentages plus addition, and the amount of money owed would then be determined using reduction. This way, students can also understand how mathematical calculations and concepts such as percentages are helpful in the real world. 7. Creating Halloween Candy Wraps In this real-world math strategy, teachers can ask students to make graphs showing their preferred candies from various brands. Young students may create life-size candy bar charts by outlining the x- and y-axes on the floor with adhesive tape and using the chocolate as bars. Older students can make tally charts and sheet graphs and observe which brands of chocolates are expensive or are more significant in size/quantity. For instance, the graph shows the price of chocolates increasing or decreasing regardless of size. Through this strategy, teachers can make students realize that mathematics applies to gardening. Students understand that they must use numbers, whether they count seeds, measure soil & seed depth, or how much water is required for planting. Through this strategy, teachers can make students realize that mathematics applies to gardening. Through landscaping, teachers can make students understand that determining the floor space area of a square or rectangular land requires multiplying the length by the width. Also, they will understand the importance of units used to measure quantities. 9. Balancing The Cheque Book It is one of the best real-world math strategies for math teachers to make students understand the use of math in managing accounts. Students can learn that financing and math go hand-in-hand. Teachers can ask the students to help their parents balance expenses and savings for a particular month. They can start by totaling all monthly debit and credit card purchases and comparing the results with the financial statements. Students can also check how much they’ve already paid in tuition fees and how much more(remaining) is to be paid next semester. In this way, students also can understand that financing and math go hand-in-hand. 10. Calculating Distances Teachers can ask students to calculate the distance and time when traveling or riding a bicycle back home, which can help students think about math concepts such as time and distance. Students can measure and share the distance they traveled from their homes to the school and figure out who lives closest and farthest from the school, and this will help them understand the concept of time and distance easily. Also Read: 10 Personalized Learning Strategies To Implement In Class Therefore, these ten real-world math strategies are great for making students understand how math concepts are related to the real world. Students can easily understand the applications and utility of mathematics through these real-world math strategies. If you are looking for some more student engagement strategies for your middle school math classroom. These real-life math strategies are easy to explain, and the things required to explain these are readily available. This will help your students relate math to their everyday life and create curiosity about the solution to the given math problem. Use these strategies with your students in your next math class and see them enjoying solving challenging math problems Free, standards-aligned math worksheets Enter your email and we’ll send you samples of our most popular math worksheets.

Solve the equation. check the solution. There are a variety of methods that can be used to Solve the equation. check the solution.. Math can be difficult for some students, but with the right tools, it can be conquered. Solving the equation. check the solution. This can help the student to understand the problem and how to Solve the equation. check the solution.. Algebra is one of the most difficult subjects for high school students. It can be very confusing, and it often involves memorizing a lot of formulas. The good news is that it doesn’t have to be this way! There are a lot of different ways you can solve algebra problems, and you can learn them all. When you learn another way to solve an algebra problem, you'll be able to see the math behind it. You’ll also understand why algebra works in the first place, which will make it easier to remember later on. By doing this, you'll be able to start solving algebra problems more easily and quickly. This will help you get better grades and make learning math less stressful. So, how do you solve algebra? First off, you want to practice by doing lots of practicing. Once you know how to solve an algebra problem, it will become easier for you to do so in the future. Second, you need to understand the concept behind it. If you don’t understand why something works in the first place, then it's going to be much harder for you to remember how to use that same method in the future. Finally, you need to identify your strengths and weaknesses when it comes to solving algebra problems. This will allow you to focus on what you're good at so that you can get better grades in the future! So, Primary school teachers can use geometry word problem solver to support their teaching by allowing pupils to explore different geometric shapes and figures. Students can choose from a range of tools and resources such as cubes, spheres, cones and pyramids to solve maths problems such as finding the area of a rectangle or calculating the volume of a cube. Modern technology has made it easier for children to learn basic maths concepts. This includes the ability to access web-based resources such as geometry word problem solver. Exponents with variables can be quite confusing. When you multiply two numbers whose exponents are both variable, you get a result that is also variable. For example, let's say you have the variable x, and the number y = 6x + 5. In this case, the exponent of y is variable because x is a variable. Now let's say you want to solve for y because you know that the exponent of y is 4. How do you solve this problem? You would factor out the variable x from both sides of the equation and find 4y = 4x + 1. This gives you the answer for y because now you know that 4y = 4(x + 1) = 4x –1. When this happens, we say that there is an "intractable" relationship between the variables on one side of an equation when they cannot be separated. An example of an equation is 3 + 4 = 7. Two numbers are added (3), then subtracted (4). This yields the solution 7. In addition to equations, there are also word problems, which require you to fill in the blanks instead of just plugging in numbers. For example, if you’re given the number $40 and asked to find 40% of the total, this is a word problem because you don’t know what “of the total” means. To solve a math problem, you need to understand how to calculate different kinds of numbers and how to read equations and word problems correctly. Lots of practice will help you get used to these techniques. They are used primarily in science and engineering, although they are also sometimes used for business and economics. They can be used to find the minimum or maximum value of an expression, find a root of a function, find the maximum value of an array, etc. The most common use of a quaratic equation solver is to solve a set of simultaneous linear equations. In this case, the user enters two equations into the program and it will output the solution (either via manual calculation or by generating one of several automatic methods). A quaratic equation solver can also be used to solve any other system of equations with fewer than three variables (for example, it could be used to solve an entire system of four equations). Quaratic equation solvers are very flexible; they can be programmed to perform nearly any type of calculation that can be done with algebraic formulas. They can also be adapted for specific applications; for example, a commercial quaratic equation solver can usually be modified to calculate electricity usage. Amazing app I understand how to solve all my math equations this app explains how to solve the equation more than my teacher there is one thing I don’t like that this app doesn’t solve geometry problems I love the app it's so helpful, if I get confused on a problem, I'll use the app to see where I messed up, I love this app though teachers aren't too fond of it because it "gives you the answers" but it helped me learn what I was taught oh it works better on the newer phones and iPods btw it may a but glitchy or leggy on the older versions but it still works pretty well

We introduce the matching measure of a finite graph as the uniform distribution on the roots of the matching polynomial of the graph. We analyze the asymptotic behavior of the matching measure for graph sequences with bounded degree. A graph parameter is said to be estimable if it converges along every Benjamini–Schramm convergent sparse graph sequence. We prove that the normalized logarithm of the number of matchings is estimable. We also show that the analogous statement for perfect matchings already fails for $d$–regular bipartite graphs for any fixed $d\ge 3$. The latter result relies on analyzing the probability that a randomly chosen perfect matching contains a particular edge. However, for any sequence of $d$–regular bipartite graphs converging to the $d$–regular tree, we prove that the normalized logarithm of the number of perfect matchings converges. This applies to random $d$–regular bipartite graphs. We show that the limit equals the exponent in Schrijver’s lower bound on the number of perfect matchings. Our analytic approach also yields a short proof for the Nguyen–Onak (also Elek–Lippner) theorem saying that the matching ratio is estimable. In fact, we prove the slightly stronger result that the independence ratio is estimable for claw-free graphs. - Miklós Abért and Tamás Hubai, Benjamini-Schramm convergence and the distribution of chromatic roots for sparse graphs, Combinatorica 35 (2015), no. 2, 127–151. MR 3347464, DOI 10.1007/s00493-014-3066-7 - M. Abért, A. Thom and B. Virág, Benjamini-Schramm convergence and pointwise convergence of the spectral measure, preprint at http://www.math.uni-leipzig.de/MI/thom/ - Itai Benjamini and Oded Schramm, Recurrence of distributional limits of finite planar graphs, Electron. J. Probab. 6 (2001), no. 23, 13. MR 1873300, DOI 10.1214/EJP.v6-96 - Béla Bollobás, The independence ratio of regular graphs, Proc. Amer. Math. 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Frenkel - Affiliation: Department of Algebra and Number Theory, Eötvös Loránd University, Pázmány Péter sétány 1/C, H-1117 Budapest, Hungary – and – Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, 13-15 Reáltanoda u., 1053 Budapest, Hungary - MR Author ID: 623969 - Email: email@example.com - Gábor Kun - Affiliation: Department of Computer Science, Eötvös Loránd University, Pázmány Péter sétány 1/C, H-1117 Budapest, Hungary - Address at time of publication: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, 13-15 Reáltanoda u., 1053 Budapest, Hungary - Email: firstname.lastname@example.org, email@example.com - Received by editor(s): January 6, 2014 - Received by editor(s) in revised form: April 15, 2014 - Published electronically: October 5, 2015 - Additional Notes: The first and third authors were partially supported by ERC Consolidator Grant 648017. The first three authors were partially supported by the Hungarian National Foundation for Scientific Research (OTKA), grant no. K109684. The second author was partially supported by the National Science Foundation under grant no. DMS-1500219 and the Hungarian National Foundation for Scientific Research (OTKA), grant no. K81310. The fourth author was partially supported by the Hungarian National Foundation for Scientific Research (OTKA), grant No. PD 109731, by the János Bolyai Scholarship of the Hungarian Academy of Sciences, by the Marie Curie IIF Fellowship, grant No. 627476, and by ERC Advanced Research, grant No. 227701. All authors were partially supported by MTA Rényi “Lendület” Groups and Graphs Research Group. - © Copyright 2015 American Mathematical Society - Journal: Trans. Amer. Math. Soc. 368 (2016), 4197-4218 - MSC (2010): Primary 05C70; Secondary 05C31, 05C69 - DOI: https://doi.org/10.1090/tran/6464 - MathSciNet review: 3453369

Mellin Transform and Image Charge Method for Dielectric Sphere in an Electrolyte We revisit the image charge method for the Green’s function problem of the Poisson-Boltzmann equation for a dielectric sphere immersed in ionic solutions. Using finite Mellin transformation, we represent the reaction potential due to a source charge inside the sphere in terms of one dimensional distribution of image charges. The image charges are generically composed of a point image at the Kelvin point and a line image extending from the Kelvin point to infinity with an oscillatory line charge strength. We further develop an efficient and accurate algorithm for discretization of the line image using Padé approximation and finite fraction expansion. Finally we illustrate the power of our method by applying it in a multiscale reaction-field Monte Carlo simulation of monovalent electrolytes. oisson-Boltzmann equation, finite Mellin transform, Green’s function, Multiscale reaction field model, Fast algorithm, Padé approximation, Finite fraction expansion 35J08; 65R10; 78A35; 82D15 The method of image charges is a classical technique [40, 23] for electrostatic problems. Its most elementary application is the problem of a point charge in a spherical cavity inside a conducting medium (or the reciprocal problem of a point charge outside a conducting sphere). In 1845, William Thomson (Lord Kelvin) noticed that the vanishing-potential boundary condition of conductors can be automatically satisfied on the sphere by putting an image point charge at the Kelvin point. A natural extension is the problem of a dielectric sphere inside a different dielectric background, where a single point image charge no longer works. In 1883 Carl Neumann discovered that a point image at the Kelvin point together with a line image starting from the Kelvin point to infinity solves the boundary condition. This result has been independently re-derived by several authors, up to 1990s; see the reviews [37, 45] for more details. More recently, the problem of a spherical cavity inside an ionic solution has been studied [8, 46]. It is again found that the image charge distribution consists of a point image and a line image. Computational method for the line charge density has been developed which works well in the asymptotic limits [8, 46] that is either large or small, where is the inverse Debye length, while is the radius of spherical cavity. The current work addresses the general case where is neither large nor small. The main advantage of image charge methods is to represent the effects of polarization charges in terms of point charges or line of charges, and therefore avoiding the task of numerically solving boundary value problems. This can substantially reduce the computational cost in Monte Carlo or molecular dynamics simulation of charged systems. Further reduction of computational cost can be achieved by discretizing line images using Gaussian quadratures, which effectively approximates a line charge by a few point charges. The total electrostatic energy of the system can then be represented as Coulomb interaction between many point charges. Most interfaces appearing in nature have irregular shapes, and the corresponding Green’s functions can not be represented in terms of simple distributions of image charges. Hence one may rightfully argue that image charge methods have rather limited applications. The most important usage of image charge methods, however, arises in the multiscale reaction-field modeling of Coulomb many body systems, where the interfaces are artificially introduced. Spherical interfaces are almost always used for their simplicity, and for the availability of analytic results for image charges. Research along this direction is of current interest in both simulations and continuum modeling [36, 12, 31, 15, 42, 5, 44]. Due to the long range nature of Coulomb interaction, simulation of charged systems is highly nontrivial. A proper treatment of boundary conditions is vital in order to obtain physically meaningful results. Periodic boundary conditions can remove artificial boundary effects in a self-consistent fashion and restore the translation symmetry. Combined with Ewald summation method , the cost of computing the total energy of the system with particles scale as . This can be further reduced to the order of using a mesh-based algorithm such as the particle mesh Ewald or particle-particle particle-mesh Ewald lattice summation techniques. The periodic images are however unphysical and may produce artifacts that obscure the real physics. Besides, computational cost of Ewald-type summation method is still rather prohibitive for large systems in Monte Carlo simulations, which limits simulation of charged systems to rather small size. The development of non-Eward methods remains important topics; See Fukuda and Nakamura for a recent review. An attractive alternative is to use the reaction field type of modeling, which is essentially a multi-scale strategy, schematically illustrated in Figure 1. In this approach, an artificial (spherical) cavity is introduced. All ions inside the cavity, together with possible mesoscopic objects such as charged proteins and colloids, are treated explicitly using microscopic model (such as the primitive model ) together with Monte Carlo/Molecular Dynamics 111Strictly speaking, grand canonical ensemble must be used in order to treat charge fluctuations properly. , while ions outside the cavity are treated implicitly using appropriate continuum theory, such as linearized Poisson-Boltzmann (PB) theory. It is known that linearized PB provides an accurate approximation to dilute electrolytes. For any charge inside the cavity, then, we must solve the electrostatic Green’s function problem, where the potential satisfies Poisson equation inside the cavity, and satisfies the (linearized) PB equation outside. The resulted Green’s function provides the pairwise electrostatic interaction (the force field) between mobile ions. This problem can be efficiently solved using the image charge method developed in this work. In this multi-scale modeling approach, the microscopic model inside the system and the continuous theory outside the cavity really describe the same system. Therefore the parameters of the Poisson-Boltzmann theory need to be determined self-consistently. These include the Debye length and the effective dielectric constant. For dilute electrolyte, the Debye length can be theoretically calculated as a function of ion densities, while the dielectric constant can be taken to be that of the solvent. Some unphysical artifacts also arise because of the artificial hard wall repulsion of the cavity surface. Using statistical mechanics, one can study this artifact and use extra short range interaction to compensate it. Alternatively, one can also ignore the details inside a thin shell near the cavity surface of thickness 1 - 2 ion diameters. Finally, to achieve the balance between efficiency and precision, the radius of cavity should be chosen to be couple of the Debye length. We also note that there can be different levels of modeling inside the cavity. In the so-called hybrid implicit/explicit model , both solvent molecules and ions are treated explicitly inside the cavity. By contrast, in the so-called hybrid primitive/implicit model, the solvent inside the cavity is modeled implicitly as a dielectric medium, while the ions are treated at the level of primitive model. The reaction field model augmented by our image charge methods is therefore able to provide an accurate treatment of the electrostatic boundary conditions. Its computation cost of the total energy scales as , where here is the total number of particles including all ions as well as their images. While for small systems (with total particle number typically smaller than 1000) this cost is manageable, for large systems, it becomes unrealistic. Fortunately, the computation cost can be dramatically reduced using fast multipole based methods [21, 4, 10, 22]. The computational cost for the combined method generally scales as . In this paper, we mainly focus on the image charge method for the Green’s function problem of these multi-scale reaction field models. The statistical mechanical foundation of these reaction field models will be discussed in a separate publication. In the remaining of this work, we shall first define the Kirkwood series for the Green’s function (Sec. II) and then use the inverse Mellin transform to find the image charge representation (Sec. III). We further discretize the line image using method of Gauss quadrature and construct an efficient numerical scheme for the computation of the reaction potential. In Sec. IV, we compute the reaction potential using our method and quantify the errors. We also demonstrate the power of multi-scale modeling by Monte Carlo simulating a dilute symmetric electrolyte. 2 Green’s function of the Poisson-Boltzmann equation As illustrated in Figure 2, let be a spherical cavity with radius , centered at the origin, with dielectric constant . The volume outside the cavity is filled with electrolyte, which is described by linearized Poisson-Boltzmann theory and is characterized by a Debye length and dielectric constant . Consider a point unit charge fixed at inside the cavity, the average potential (averaged over the statistical fluctuations of electrolyte outside the cavity) satisfies the following equations: where and are the average potential inside and outside the cavity respectively. The boundary condition at infinity is Throughout the paper we use light italic letter to represent the magnitude of a vector . On the cavity boundary, the Green’s function satisfies the following standard electrostatic interface conditions: The general boundary value problem associated with Eqs. (1)-(3) actually defines the electrostatic Green’s function, , which equals inside the cavity and outside. It is this Green’s function that shall be directly used in the multi-scale reaction field modeling of electrolyte. The inverse Debye length is defined by where is the Bjerrum length of the solvent ( for water at room temperature), and are the bulk concentration and the valence of the th species of ions. The Green’s function defined in Eqs. (1)-(3) has an azimuthal symmetry, hence depends only on and , where is the angle between the source point and the field point , see Fig. 2. The potential inside the cavity can be written as, which is the superposition of the direct Coulomb potential and the reaction potential which is a harmonic function. Both potentials can be expanded in terms of spherical harmonics: |where is the smaller (larger) one between and , while are constants to be determined, and is Legendre polynomial of order . The potential outside the cavity can also be expanded in terms of Legendre polynomials:| where is the modified spherical Hankel function (also called the modified spherical Bessel function of the third kind) , defined by the following series, For source charge inside the cavity, we always have Using the orthogonality of the spherical harmonics, we obtain the following expansion of the reaction potential, widely known as Kirkwood series expansion : with the harmonic coefficients given by where and The Kirkwood series expansion has been used in the calculation of the reaction potential inside a spherical cavity in simulations [24, 6]. It converges slowly when a source charge approaches to the cavity surface, which prevents its wide application in dynamical and Monte Carlo simulations. 3 Image charge representation and algorithm In this section, we develop an image charge representation for the reaction potential using finite Mellin transformation. 3.1 Finite Mellin transform Let be a function defined in the interval . Its finite Mellin transform, is defined as where is generically a complex variable. The original function can be expressed in terms of by the inverse Mellin transform Here the integration is carried out over a vertical line, , in the complex plane. The dual functions form a finite Mellin transform pair. The finite Mellin transform is closely related to the one-sided Laplace transform. Let , then Likewise, the inverse transform can also be expressed in terms of the inverse Laplace transform. The finite Mellin transform is equivalent to the usual Mellin transform for the function with compact support in the finite interval. The usual Mellin transform can be defined in terms of the two-sided Laplace transform. Two finite Mellin transform pairs shall be useful in our discussion below, where is the Dirac delta function. The Mellin transformation of the second pair is defined for . We shall consider in Eq. (10) as a function of and analytically continue the integer variable into the complex plane. This can be done through using the integral representation of Bessel functions, for which and the modified Bessel function of the second kind (see, e.g. Ref. , pp. 917) for . Let be the inverse finite Mellin transformation of ,222 We shall suppress the dependence of on , to avoid cluttered notations. we have As given by Eq. (10), the harmonic coefficient is finite for all , which ensures the existence of the finite Mellin transform. Substituting Eq. (15b) into the Kirkwood series Eq. (9), and using , the reaction potential can be re-expressed as Let us further define a vector . As decreases from to , the vector runs from the Kelvin point to infinity along the radial direction. We shall see that this is precisely the loci of the image charge line. Let be the magnitude of vector , we have . Since we are only interested in the field point inside the cavity, we have , hence we can sum the series in Eq. (16) using the expansion Eq. (5a) and obtain The last integral represents the potential generated by one dimensional distribution of image charges along the radial direction, which starts from the Kelvin point and extends to infinity. The linear charge density is . The geometry is illustrated in Fig. 2. Therefore we arrive at our main result in this work. where the line charge density is and . The image charge representation for the reaction potential Eq. (18) is equivalent to the Kirkwood series representation, Eq. (9). It is advantageous because the line integral can be efficiently discretized by Gauss quadrature. A few Gauss points can provide approximations with accuracy as high as desired . Physically, this amounts to approximating line image by a few point images. 3.2 Point image and line image The following limit of the Bessel function can be established (using Eq. (28)) Combining with Eq. (10) we obtain the large limit of the coefficients : The coefficients Eq. (10) can therefore be decomposed into two parts where vanishes as goes to infinity. The inverse Mellin transform of a constant is a delta function , while the inverse Mellin transform of the function is generally a continuous function in the interval . The linear charge density therefore can be decomposed into (with ): The first term corresponds to a point image at the Kelvin point, and the second term corresponds to a continuous line image extending from the Kelvin point to infinity. The simplest limit is , where the exterior of the cavity becomes a conductor. In this limit, we easily see from Eq. (10) that , and . The image charge distribution reduces to a single point image at the Kelvin point, a well known result. When the dielectric constants inside and outside the cavity are the same, i.e. , the point image vanishes, but the line image persists. This is precisely the case of multi-scale reaction field model for electrolytes. 3.3 Neumann’s result revisited Let us review the simple case where there is no screening ions outside the cavity, i.e. . In 1883, Neumann found an exact expression for the reaction field, in terms of a point image charge at the Kelvin point and a line image: where . This formula has been re-derived independently by various authors in different fields of applications [48, 14, 11, 38, 28, 35]; also see Lindell’s review for the summary of history. The Mellin transform method was also used by Lindell and collaborators [29, 34] for finding image charges of the Poisson equation in layered media. Its inverse Mellin transform can be exactly calculated using Eq. (13): The application of the Neumann’s result in molecular dynamics can be found in two recent papers [26, 27]. The reciprocal problem of a source charge placed outside of a sphere was also applied in Monte Carlo simulations of colloidal systems [9, 19]. Historically, what is widely used in computer simulations of biological systems is only single image charge approximations, see works by Friedman and Abagyan and Totrov . These methods are of the first or second order accuracy in the dielectric ratio . They fail to be accurate if the ratio is not small. 3.4 Limits of large cavity and small cavity For , the inverse Mellin transform of can not be exactly calculated. One way to proceed is to use the following asymptotic approximation , which gives the correct leading order asymptotics both for and for . Substituting it back into Eq. (10) leads to where . The inverse Mellin transform of this can be easily found. This approximation works well for the case of small (small cavity) and large (large cavity). 3.5 Large asymptotics of Let us first look at the large limit of the Bessel function . For sufficiently large , the largest term in the sum Eq. (6) is given by . We can therefore rewrite the summation as where we have defined . We can expand the function being summed into asymptotic series in terms of and extend the upper limit of summation from to . The resulting summation then can be calculated order by order in . This can be conveniently done using Wolfram Mathematica. For example, up to order of , we have It then follows that Generically, therefore, the leading order term of scales as . In the most interesting case , however, this term vanishes and we have Now consider the limit where the source charge approaches the cavity boundary, we have . The reaction potential acting on the source charge then (see Eq. (9)) becomes Therefore the line image strength must vanish at the Kelvin point. 3.6 The general case 3.6.1 Padé approximations to harmonic coefficients In the multi-scale reaction field model, we typically have and of order of unity. All methods discussed above fail in this case. To obtain accurate approximation, we approximate the harmonic coefficients by a rational function of (i.e. Padé approximation): with constants and for all . Clearly Eq. (25) is a special case of Eq. (32) with . The second term on the right hand side is the order Padé approximant 333Here and refer to the degree of polynomials in the numerator and in the denominator respectively. of the function , and the rational polynomial preserves the asymptotics, The coefficients are solved by the nonlinear least square method. For given and dielectric ratio , the constants in the expansion can be simply determined by a minimization of the total error of the first terms of the harmonic coefficients, The Newton iteration scheme or other iteration algorithms can be applied to solve this nonlinear optimization problem. To demonstrate the quality of Padé approximation, we list in Table 1 the relative errors for the cases of and 3 with various parameters and : where is the numerical solution of the nonlinear least square problem (34). We take since 51 multipoles in the Kirkwood series already provide sufficiently high accuracy. It should be pointed out that the minimization solution depends on the initials and may be not a global minimization. Nonetheless, the results in Table 1 clearly show the remarkable precision of the Padé approximation. For example, the worst case for has a relative error . In comparison, the asymptotic method Eq. (25) yields much larger error, and completely breaks down for . |Asymptotics Eq. (25)| 3.6.2 Image expressions through inverse Mellin transforms After the best-fitting coefficients are determined, we can further re-express the Padé approximation Eq. (32) using the partial fraction expansion: where the coefficients and can be determined by the Heaviside’s cover-up method. Using Eq. (13), the inverse Mellin transform , we easily find the image charge distribution: with represents a line image density from the Kelvin image point to the infinity along the radial direction. Alternatively, the reaction potential is, Figure 3 illustrates the profiles of line image strengths for and for varying from to 1 calculated with . Note that the line image is always oscillatory, which implies that some of the parameters are complex numbers. Note also that both amplitude and period increase with the dielectric ratio . 3.6.3 Discretization of the line image Computation of the line integral in Eq. (38) using the continuous linear charge density Eq. (37) is still expensive. This is a quite severe limitation on the computational efficiency, since the line integral has to be computed in every simulation step. Therefore we further discretize the line integral (38) using numerical quadrature. This amounts to approximating the linear image using multiple point image charges. An efficient discretization scheme uses fewer point images, and for large-scale systems the computation of pairwise interactions of these source-image charges can be speeded up with fast multipole-type algorithms [4, 10, 22, 47] to achieve approximately linear complexity, significantly reducing the computational cost in computer simulations. We consider the case of . Extension of the algorithm to other values of is straightforward. Since must be real and the line charge is oscillatory, the parameters in Eq. (35) are generally composed of a positive real number and a complex conjugate pair. The same clearly also holds for the set of parameters . Let be real and The line image strength in Eq. (37) can be expressed as where . Note that the second term is oscillatory, in agreement with Fig. 3. Approximating an oscillating integral is tricky. We divide the integral in Eq. (38) into two parts: with a positive number sufficiently larger than unity. The first integral in Eq. (41) can be transformed into an integral over the interval through the linear variable transformation: with and a positive constant. Due to the finite interval being integrated, the integrand is only weakly oscillatory. Further defining a function via we can easily show that The resulting integral over can be then integrated using the classical -point Gauss-Legendre quadrature, which discretizes the line image into several point image charges and are the Gauss quadrature points and weights. For the second integral in the RHS of Eq. (41), the Gauss quadrature is less efficient due to the high oscillatory integrand for small . Fortunately, the variable is much larger than , hence the integrand can be expanded in terms of the ratio : The first few terms can be explicitly worked out using Eq. (40): In summary, the reaction potential Eq. (38) is approximated by image point charges plus a few correction terms, where the term of represents the Kelvin image charge, with and . As pointed out previously, the Kelvin image vanishes when . Remark. The error of approximating Eq. (41) comes from two sources. One is from the Gauss quadrature to the finite integral, and the other is from the truncation of multipoles for the infinite integral. Both errors depend on the cutoff parameter . In approximating infinite integral, the leading term in the error of the truncation is where . The Gauss quadrature has a fast convergence, hence we expect a small such that a few points leading to high accuracy. We find provides a good balance between two approximations in an accuracy of 3 digits. 4 Numerical results In this section, we compute the reaction potential using our method and quantify the errors. We also demonstrate the power of multi-scale modeling by Monte Carlo simulating a dilute symmetric electrolyte. 4.1 Accuracy and efficiency of discretization scheme We first test the accuracy of our discretization scheme. We focus on the most difficult case where the line image is strongly oscillatory, and the point image vanishes. It is also the case appears in the multiscale modeling of electrolytes. We take and , and calculate the relative error of the self energy of a unit charge, , as a function of the source point radius by comparing with 201-terms-truncation of the Kirkwood series. The latter has a truncation error about (for example, when the error is about ) and can be considered as the “exact” solution. We set and . We find that the accuracy of numerical quadrature only weakly depends on and . We use and Gauss quadrature points for the integration on the finite interval and the two different corrections for the integration on interval : and . Note that means the second term of Eq. (47) is a constant correction. The results are shown in Figure 4. It is seen that the image charge approximation is very accurate even with only 4 image points, with the overall relative errors remaining less than . The accuracy however does decrease when approaches to the boundary. To determine the asymptotics of image charge approximation for the self energy near the boundary, we also compare it with the direct truncation of the Kirkwood series, in the range . These results, shown in Figure 5, clearly demonstrate that image charge approximation converges much faster than the Kirkwood series near the boundary. In particular, image charge approximation with 4 image charges is uniformly better than Kirkwood series with 20 terms. For example, the error of Kirkwood series with 20 terms is larger than for , and increases to for , while the error of the image charge approximation remains less than . We also compare the CPU timing efficiency of our image charge method and that of the Kirkwood series. The comparison was performed using the optimized module for the Bessel functions by Matlab, and the algorithm of the Legendre polynomials in Numerical Recipes . The machine used has double-core 2.3GHz CPU and 8G memory. We use the same system parameters as in the accuracy test and calculate the pairwise energy of ions randomly distributed in the cavity. In the image method we use images and and 1 corrections, respectively. The Kirkwood series are truncated at 10th, 20th and 50th terms. The results are listed in Table 2. We see the image method with 4-point images is generally 40-50 times faster than the method of Kirkwood series truncated at 20-th terms. With these parameters, two methods have similar accuracy. 4.2 Application in Monte Carlo simulations of electrolytes To illustrate the utility of our image charge method, we apply it in a reaction-field Monte Carlo simulation of electrolyte. We artificially introduce a spherical cavity with radius in the electrolyte, and model all charges in the cavity using the primitive model and simulate them using Monte Carlo method. All ions outside the cavity, together with the solvent, are treated implicitly using linearized PB theory, characterized by two parameters: the dielectric constant and the inverse Debye length . For any ion inside the cavity, the effects of the electrolyte outside the cavity is to introduce a reaction potential, which can be calculated using our image charge method. It is important to note that the inverse Debye length characterizing the medium outside the cavity shall be self-consistently determined by the simulation of the ions inside the cavity. We run canonical Monte Carlo simulations of the primitive model using the standard Metropolis criterion [32, 16] for particle displacements. The ions are modeled by hard spheres with diameter and with a point charge of valence at its center. The ions are mobile in the solvent medium with dielectric permittivity . The effective Hamiltonian of the system is given by The self energy is given by where is the Bjerrum length and is the Boltzmann factor. We take for the water permittivity at room temperature. The infinite potential is due to the presence of the hard wall at The pairwise interaction energy is given by As is well known, the reaction potential is symmetric under permutation of two variables: . We first calculate the density distribution by taking a system with and the salt concentration is which corresponds to a Debye length . We calculate both the cases with and without the reaction field. In the former case, we use different numbers of image charges varying from 4 to 6, with the parameters for numerical quadratures the same as those in the upper panel of Figure 4. For each setting, we run MC cycles for each particle to obtain samples for the statistics of particle number in each spherical shell with thickness . In Figure 6 we show the radial distribution function of anions, i.e., the normalized density, . The distribution of cations is similar. Evidently, without accounting for the reaction field, the density is higher near the center of cavity and lower near the cavity boundary. The difference in the density is up to 3 percent. When the reaction field is taken into account, the particle density shows variation less than in most regions inside the cavity. There is substantial deviation of density near the cavity boundary. This is due to the presence of the artificial hard wall on the boundary, see Eq. (49). This problem can be cured by adding to the self energy a short ranged correction, or by introducing a buffer zone of a certain thichness (). The latter approach has been discussed in literature [3, 43, 26]. We shall present statistical mechanical discussion of the former approach in a separate publication. Finally, in Figure 7, we plot the results for three different bulk ionic concentrations, where and respectively. These results again show the necessity of treating the reaction potential, as well as the accuracy and efficiency of our image charge methods. It is also interesting to note that as the Debye length decreases the density becomes flatter, suggesting that the reaction-field Monte Carlo models becomes more precise. For the case of salt concentration , for example, the Debye length is approximately half of the cavity radius, while the density variation becomes less than (excluding the thin region affected by the hard wall artifacts). In summary, we have developed an image method for charges inside a spherical cavity that is immersed in an ionic solution. Our method is useful for multiscale reaction field models of electrolytes and other more complicated charged systems. We derive an analytic expression for the reaction potential in terms of one dimensional image charge distribution, and discuss a highly accurate and efficient algorithm for discretizing the image line charge. We also apply our method to a reaction field Monte Carlo simulations of electrolytes. and demonstrate the accuracy and efficiency of the new algorithm. Simulation of charged systems is computationally expensive, therefore is always limited to small system size. In a physical system of such size, the total charge may fluctuate away from zero to a noticeable extent. These fluctuations can not be taken into account in canonical ensemble simulations. In another word, grand canonical ensemble must be used to capture the charge fluctuations of small systems. Ewald-summation method, which is so far the most popular simulation methods for charged systems, are based on periodic boundary conditions, and are difficult to be incorporated with grand canonical ensemble. 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Inequality Graphing App Graphing inequalities is made simple for the beginning or intermediate calculator user with the Inequality Graphing App. Read the License before continuing. By downloading the application you indicate your agreement with the terms and conditions of the License. Subject to your payment of any applicable license fee, Texas Instruments Incorporated "TI" grants you a license to copy and use the software program s on a TI calculator and copy and use the documentation from the linked web page or CD ROM both software programs and documentation being "Licensed Materials". Systems of Equations and Inequalities In previous chapters we solved equations with one unknown or variable. We will now study methods of solving systems of equations consisting of two equations and two variables. Represent the Cartesian coordinate system and identify the origin and axes. Given an ordered pair, locate that point on the Cartesian coordinate system. Given a point on the Cartesian coordinate system, state the ordered pair associated with it. We have already used the number line on which we have represented numbers as points on a line. Note that this concept contains elements from two fields of mathematics, the line from geometry and the numbers from algebra. Rene Descartes devised a method of relating points on a plane to algebraic numbers. This scheme is called the Cartesian coordinate system for Descartes and is sometimes referred to as the rectangular coordinate system. This system is composed of two number lines that are perpendicular at their zero points. Perpendicular means that two lines are at right angles to each other. Study the diagram carefully as you note each of the following facts. The number lines are called axes. Example 3 Graph the solution for the linear inequality 2x - y ≥ 4. Solution Step 1: First graph 2x - y = 4. Since the line graph for 2x - y = 4 does not go through the origin (0,0), check that point in . Fit an algebraic two-variable inequality to its appropriate graph. If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *pfmlures.com and *pfmlures.com are unblocked. Solve, Graph and Write Inequalities Make sure that you can draw a graph from an inequality and write an inequality when given a graph. Pay attention to open and closed circles. The horizontal line is the x-axis and the vertical is the y-axis. The zero point at which they are perpendicular is called the origin. Positive is to the right and up; negative is to the left and down. The arrows indicate the number lines extend indefinitely. Thus the plane extends indefinitely in all directions. The plane is divided into four parts called quadrants. These are numbered in a counterclockwise direction starting at the upper right. Points on the plane are designated by ordered pairs of numbers written in parentheses with a comma between them, such as 5,7. This is called an ordered pair because the order in which the numbers are written is important. The ordered pair 5,7 is not the same as the ordered pair 7,5. Points are located on the plane in the following manner. First, start at the origin and count left or right the number of spaces designated by the first number of the ordered pair. Second, from the point on the x-axis given by the first number count up or down the number of spaces designated by the second number of the ordered pair. Ordered pairs are always written with x first and then y, x,y. The numbers represented by x and y are called the coordinates of the point x,y. The first number of the ordered pair always refers to the horizontal direction and the second number always refers to the vertical direction. Check each one to determine how they are located. What are the coordinates of the origin? Find several ordered pairs that make a given linear equation true. Locate these points on the Cartesian coordinate system. Draw a straight line through those points that represent the graph of this equation. A graph is a pictorial representation of numbered facts.We can also graph inequalities on the number line. The following graph represents the inequality x≤2. The dark line represents all the numbers that satisfy x≤2. If we pick any number on the dark line and plug it in for x, the inequality will be true. Write an inequality for the graph Download jpg. Ask for details ; Follow Report by Richy1 03/09/ It is d you are right because open circle mean and it was pointing to the right so that means it was x>3. 0 votes Get the Brainly App5/5(1). Solving and Graphing Absolute Value Inequalities: Practice Problems; Studying with Flashcards. If you are studying for a test on inequalities, these flashcards can provide a review of the main concepts. This Write and Graph Inequalities: Temperature Video is suitable for 6th - 8th Grade. What temperature does water freeze? Have your learners show in an inequality all the possible temperatures. The video demonstrates how to set up an inequality based on the situation given and how to graph it. This quiz and worksheet will help you assess your knowledge of writing equations with inequalities. To pass the quiz you must be familiar with signs related to writing these equations and other. Write and graph a linear inequality that represents th situation. To start out, you have to write the inequality they are asking for. I'm assuming we can't spend any more than $48, but we don't have to spend all of $ If it costs $5 for every album (x) and $8 for every movie (y), than we'd write an inequality: App for Students.

The standard deviation is one of the main tools in the analysis of the data and it is dispersion. When you are able to do the Standard deviation test by the Standard deviation calculator, then it is easy for us to analyze the whole data. When we are using the sd calculator, then we analyze what is the mean values and what are the upper and the lower limit of the whole population of the data. In reality, it is impossible to analyze the whole population of the data, and you always go for a sample evaluation. When you apply the test on the test on the sample by the standard deviation. Then the whole picture about the whole population would be quite clear for you. Calculate standard deviation and find the information about the whole population of the data, you can evaluate what is the highest values and what are the lowest values and between the lower the upper limit the whole set of the data is residing. In the following article, we are discussing what is the importance of the standard deviation and what is its utilization. Why do we use standard deviation? The standard deviation is one of the main tools in defining the depth of the data in our observation. We are going to find all the variance in data by the mean and standard deviation calculator. When we are able to find the variance in the whole data, then we are able to predict the following calculation: - The mean of the data, and the culture of the values around it. This would represent the whole population of the data and its resonating values. - The upper and the lower limit of the data and clarify the picture of how the data is dispersed and what its variance is around the whole set of the data. In the following example, we are defining the mean, variance, and standard deviation from the set of the data Question: Find the mean, variance, and standard deviation for the following data? |Class Interval||Frequency (f)||Mid Value (xi)||fxi||fxi2| |0 – 10||27||5||135||675| |10 – 20||10||15||150||2250| |20 – 30||7||25||175||4375| |30 – 40||5||35||175||6125| |40 – 50||4||45||180||8100| |50 – 60||2||55||110||6050| |?f = 55||?fxi = 925||?fxi2 = 27575| N = ?f = 55 Mean = (?fxi)/N = 925/55 = 16.818 Variance = 1/(N – 1) [?fxi2 – 1/N(?fxi)2] = 1/(55 – 1) [27575 – (1/55) (925)2] = (1/54) [27575 – 15556.8182] Standard deviation = ?variance = ?222.559 = 14.918 Analysis of the concepts: The complete analysis of the question is essential to understand the concept of the standard deviation and how we can find the standard deviation by the sample standard deviation calculator. The population standard deviation calculator can analyze the whole set of the population in a matter of seconds. But you should be familiar with the following concepts to make sure, that you are familiar with the whole result. The first and the foremost concept is the class interval, in this case, the class intervals are 0-10,10-20, 20-30,30-40, 40-50, 50-60. These are the class intervals, actually, we have divided the whole set of data into small and regular intervals to know their frequencies and how much data is residing in one specific interval. The frequencies of various intervals are 27, 10, 7, 5, and 4,2, and these are the frequencies of various class intervals. For example, the interval 0-10 has a frequency of 27 and the class 10-20 has a frequency of 10, and the 7, 5, 4, and 2 for the proceeding class intervals. The class interval 0-10 has the highest frequency and the class interval 50-60 has the minimum frequency of 2. Mid values (xi): The mid values are the middle values of our data range, for example, the class interval 0-10 has a middle range of 5. The class interval 0-20 has a middle range of 15, 25, 35, 45, and 55 for the next class intervals. The frequency of middle values( fxi): The Standard deviation calculator(fxi) can be calculated by multiplying the frequency (f) and the middle values. In the above question, the frequencies of the middle values are given as 135, 150, 175, 175,180,110. The sd calculator readily finds the frequencies of the middle values and we are easily able to evaluate the values. The total frequency of middle(fxi2): The total frequency of the middle values is calculated by the frequency of the middle values. In the above question the the total frequencies are 675, 2250, 4375, 6125, 8100, 6050 for the class intervals 0-10, 0-20, 0-30, 0-40, 0-50, 0-60.The total frequency is telling how many times, the mean values are residing in our calculations. This can be extracted from the data, and it would be greatly helpful; in finding the frequency of the data. The Summations ?f, ?fxi, ?fxi2 : The summation ?f, ?fxi, ?fxi2 is the simple addition of all the above calculations. Now the ?f is the total frequency, ?fxi is the total frequency of the middle values. When we are able to find the sample standard deviation by the simple formula if we are able to find all the calculations. The main thing for the students to learn is a concept like the Standard deviation, it is essential to learn the basics like the class interval, frequency, the mean frequency, and the total mean frequency. When you are able to extract all the data values, then it is easy for the students to calculate all the variance and the standard deviation calculation. It would be great for the students doing the research as the standard deviation is vastly used in the analysis of the data and the profile for your data.

Formulating the research question or developing the hypothesis can help you to decide on the approach of the research. A research question is the question the research study sets out to answer. Print What is a Hypothesis? A hypothesis is a tentative, testable answer to a scientific question. Once a scientist has a scientific question she is interested in, the scientist reads up to find out what is already known on the topic. Then she uses that information to form a tentative answer to her scientific question. Sometimes people refer to the tentative answer as "an educated guess. A hypothesis leads to one or more predictions that can be tested by experimenting. Predictions should include both an independent variable the factor you change in an experiment and a dependent variable the factor you observe or measure in an experiment. A single hypothesis can lead to multiple predictions, but generally, one or two predictions is enough to tackle for a science fair project. Examples of Hypotheses and Predictions Question Prediction How does the size of a dog affect how much food it eats? Larger animals of the same species expend more energy than smaller animals of the same type. To get the energy their bodies need, the larger animals eat more food. If I let a pound dog and a pound dog eat as much food as they want, then the pound dog will eat more than the pound dog. Does fertilizer make a plant grow bigger? IRENE PEPPERBERG Research Associate, Psychology, Harvard University; Author, The Alex Studies The Fallacy of Hypothesis Testing. I've begun to rethink the way we teach students to engage in scientific research. A hypothesis is an explanation for a set of observations. Here are examples of a scientific hypothesis. Although you could state a scientific hypothesis in various ways, most hypothesis are either "If, then" statements or else forms of the null hypothesis. The null hypothesis sometimes is called the. In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite urbanagricultureinitiative.com states: There is no set whose cardinality is strictly between that of the integers and the real numbers.. The continuum hypothesis was advanced by Georg Cantor in , and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in Plants need many types of nutrients to grow. Fertilizer adds those nutrients to the soil, thus allowing plants to grow more. If I add fertilizer to the soil of some tomato seedlings, but not others, then the seedlings that got fertilizer will grow taller and have more leaves than the non-fertilized ones. Does an electric motor turn faster if you increase the current? As more current flows through the motor's electromagnet, the strength of the magnetic field increases, thus turning the motor faster. If I increase the current supplied to an electric motor, then the RPMs revolutions per minute of the motor will increase. Is a classroom noisier when the teacher leaves the room? Teachers have rules about when to talk in the classroom. If they leave the classroom, the students feel free to break the rules and talk more, making the room nosier. If I measure the noise level in a classroom when a teacher is in it and when she leaves the room, then I will see that the noise level is higher when my teacher is not in my classroom. What if My Hypothesis is Wrong? What happens if, at the end of your science project, you look at the data you have collected and you realize it does not support your hypothesis? First, do not panic! The point of a science project is not to prove your hypothesis right. The point is to understand more about how the natural world works. Or, as it is sometimes put, to find out the scientific truth. When scientists do an experiment, they very often have data that shows their starting hypothesis was wrong. Well, the natural world is complex—it takes a lot of experimenting to figure out how it works—and the more explanations you test, the closer you get to figuring out the truth. For scientists, disproving a hypothesis still means they gained important information, and they can use that information to make their next hypothesis even better. In a science fair setting, judges can be just as impressed by projects that start out with a faulty hypothesis; what matters more is whether you understood your science fair project, had a well-controlled experiment, and have ideas about what you would do next to improve your project if you had more time. You can read more about a science fair judge's view on disproving your hypothesis here. It is worth noting, scientists never talk about their hypothesis being "right" or "wrong.IRENE PEPPERBERG Research Associate, Psychology, Harvard University; Author, The Alex Studies The Fallacy of Hypothesis Testing. I've begun to rethink the way we teach students to engage in scientific research. Although you could state a scientific hypothesis in various ways, most hypothesis are either "If, then" statements or else forms of the null urbanagricultureinitiative.com null hypothesis sometimes is called the "no difference" hypothesis. It only took five minutes for Gavin Schmidt to out-speculate me. Schmidt is the director of NASA ’s Goddard Institute for Space Studies (a.k.a. GISS) a world-class climate-science facility. One. When you answered this question, you formed a hypothesis. A hypothesis is a specific, testable prediction. It describes in concrete terms . Question: "What is the documentary hypothesis?" Answer: The documentary hypothesis is essentially an attempt to take the supernatural out of the Pentateuch and to deny its Mosaic authorship. The accounts of the Red Sea crossing, the manna in the wilderness, the provision of water from a solid rock, etc., are considered stories from oral tradition, thus making the miraculous happenings mere. hypothesis as, "a tentative explanation for an observation, phenomenon, or scientific problem that can be tested by further investigation." This means a hypothesis is the .

- About this Journal - Abstracting and Indexing - Aims and Scope - Annual Issues - Article Processing Charges - Articles in Press - Author Guidelines - Bibliographic Information - Citations to this Journal - Contact Information - Editorial Board - Editorial Workflow - Free eTOC Alerts - Publication Ethics - Reviewers Acknowledgment - Submit a Manuscript - Subscription Information - Table of Contents Abstract and Applied Analysis Volume 2013 (2013), Article ID 629468, 10 pages Convergence Theorems for Fixed Points of Multivalued Strictly Pseudocontractive Mappings in Hilbert Spaces 1Mathematics Institute, African University of Science and Technology, PMB 681, Garki, Abuja, Nigeria 2Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA 3Université Gaston Berger, 234 Saint Louis, Senegal 4Department of Mathematical Sciences, Bayero University, PMB 3011, Kano, Nigeria Received 10 September 2012; Accepted 15 April 2013 Academic Editor: Josef Diblík Copyright © 2013 C. 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Frederik Denef†† , Michael R. Douglas†† , Bogdan Florea†† , Antonella Grassi†† and Shamit Kachru†† Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855-0849, USA I.H.E.S., Le Bois-Marie, Bures-sur-Yvette, 91440 France Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA Department of Physics and SLAC, Stanford University, Stanford, California, CA 94305, USA We discuss a simple example of an F-theory compactification on a Calabi-Yau fourfold where background fluxes, nonperturbative effects from Euclidean D3 instantons and gauge dynamics on D7 branes allow us to fix all closed and open string moduli. We explicitly check that the known higher order corrections to the potential, which we neglect in our leading approximation, only shift the results by a small amount. In our exploration of the model, we encounter interesting new phenomena, including examples of transitions where D7 branes absorb O3 planes, while changing topology to preserve the net D3 charge. All well-studied quasi-realistic string compactifications come, in the leading approximation, with large numbers of scalar moduli fields. It has long been thought that this is an artifact of the leading approximation, and that in many backgrounds non-perturbative effects generate potentials which lift all of these fields. More recently, it has been emphasized that such potentials can have a very large number of minima, which combined with other choices made in the constructions gives rise to an immense ‘landscape’ of string vacua [22,77,1,65,96,43]. Indeed, very simple genericity arguments, based on knowledge of effective field theory and of the contributions of various stringy ingredients to the effective potential, suggest that this should be true. Since typical constructions have dozens or hundreds of moduli fields, and many different effects contributing to the scalar potential, the explicit calculations required to verify this expectation are daunting in almost all examples. However it is important to pursue explicit examples in as much detail as possible. Such examples allow one both to verify the hypothesis that there are many models with stabilized moduli, and to gain further intuition about stringy potentials which may be relevant in developing models of inflation or particle physics. A proposal for stabilizing a wide class of type IIB or F-theory compactifications was put forth in , and fairly explicit examples were provided in . In F-theory on an elliptic Calabi-Yau fourfold, the moduli of interest are the complex structure moduli and the Kähler moduli (which come from of the base of the elliptic fibration). In the limit where one can think of these constructions in terms of IIB Calabi-Yau orientifolds, these moduli include the moduli of the Calabi-Yau threefold, the axio-dilaton, and the positions of D7-branes in the threefold geometry. In the constructions described in , one approaches the problem of moduli stabilization in two steps. First, one turns on background fluxes to stabilize the complex structure moduli of the fourfold at some energy scale – the fluxes can in general stabilize both the complex structure and D7 brane moduli, from the IIB perspective. Then, one incorporates exponentially small effects arising from Euclidean D3 instantons (corresponding to M5 instantons wrapping vertical divisors of holomorphic Euler characteristic†† This is often (but not quite accurately) called the arithmetic genus in the physics literature in this context. ) or infrared gauge dynamics on D7 branes, to generate a potential for the Kähler moduli at the scale . At any moderately large radius in string units, one finds , and it is a good approximation to treat the complex moduli as fixed when one evaluates the effects generating a potential for the Kähler modes (so any complex structure-dependent determinants multiplying the instanton action, should be evaluated at the critical point of the flux potential). As described in , as long as the gravitino mass resulting from the first step is moderately small in string units, one has a small parameter to use in the next step, which can result in radii stabilized within the regime of convergence of the instanton expansion. In fact, as we shall see explicitly, even values of can suffice in some concrete models. In this paper, we provide an explicit example where this recipe is carried out in complete detail. We construct an orientifold of a Calabi-Yau threefold and its F-theory dual fourfold, which contains enough rigid divisors and pure non-abelian D7 brane gauge theories, to generate a potential for all Kähler modes. The Calabi-Yau threefold is the resolved orbifold , which has 51 Kähler moduli and 3 complex structure moduli. It is not difficult to construct explicit flux vacua in the complex structure sector of this model, and we provide some examples with moderately small . We solve for the stabilized values of all 51 Kähler moduli in the leading approximation, and show that the subleading corrections which we neglected are indeed expected to be quite small. Our example was chosen for its simplicity, especially as regards its complex structure moduli space (since we wished to provide explicit flux vacua), and hence it admits far fewer flux vacua than more generic fourfolds – only , in contrast to the more impressive numbers like that arise in more complicated examples. In suitable cases with larger numbers of flux vacua, one expects similar constructions to be even more controlled, but of course the higher-dimensional complex structure moduli space becomes harder to work with explicitly. The organization of this paper is as follows. In §2, we give a first description of our main model. We find that the Kähler moduli space has a rather intricate structure, with different phases that meet in a singular orbifold geometry. We argue that each blow-up mode of the orbifold which contributes a twisted Kähler mode in the theory, comes along with its own vertical divisor of holomorphic Euler characteristic one, and that the gauge theory sector is pure SYM without any matter. In §3, we describe the geometry of the F/M-theory dual fourfold in great detail, and put the arguments of §2 on firm mathematical footing. In §4, we discuss the relation of our model to some similar toroidal orientifolds which have (in absence of flux) exactly soluble worldsheet definitions at the orbifold point. In §5, we construct explicit flux vacua in this example, which stabilize the complex structure moduli and provide the needed moderately small . We choose to saturate the entire D3-tadpole by turning on fluxes, so no wandering D3 branes are introduced in the geometry. In §6, we then include the leading non-perturbative effects, and see that once one includes both the D3-instanton effects and the gauge dynamics on the D7 branes in this model, all untwisted and twisted Kähler moduli are stabilized. We check the magnitude of the known corrections to the leading result, from perturbative and worldsheet instanton corrections to the Kähler potential as well as multi-instanton contributions to the superpotential, and see that even with our only moderately small these corrections are expected to be at the one percent level. In the penultimate section, we briefly describe some other explicit models with only a single Kähler mode which are good candidates for stabilization. We conclude in §8. In the appendix, we describe our normalization conventions. 2. Orientifold Geometry Our model is a type IIB orientifold of a smooth Calabi-Yau threefold , which is a resolution of the orbifold . The orbifold group acts as There are fixed lines under the action of a group element, and fixed points where three fixed lines meet. Blowing up the fixed lines resolves the geometry and introduces 48 twisted Kähler moduli in addition to the 3 untwisted Kähler moduli descending from . There are just 3 complex structure moduli, given by the modular parameters of the factors. Thus the resulting smooth Calabi-Yau threefold has , . This is one of the Borcea-Voisin models [21,100]. We will see in the following that we can choose an orientifold involution of and D7-brane embeddings in such way that we get an gauge group without matter, and a D3 tadpole (on the quotient space) induced by non-exotic O3-planes and 7-brane curvature. Before proceeding, let us briefly explain why we do not define our model as a toroidal orientifold, i.e. by orientifolding the orbifold . Despite the geometric singularities, orientifolds of an orbifold can often be described directly by perturbative string CFT’s, and the case of has been studied extensively. At first sight, the toroidal orientifold appears promising: the involution , which (modulo the equivalences) has O7 fixed planes and 64 fixed points at the triple intersections of the O7 planes, each of which corresponds to an O3 plane, roughly matching the smooth geometry we are about to describe. This comparison will be carried much further in §4, where it is shown that the gauge group and many other features of our model can be realized by a toroidal orientifold. However, in general, not every large radius orientifold model has an orbifold limit with a well-behaved CFT description. When trying to go continuously from large radii to the orbifold point, nonperturbative massless states could arise from D-branes wrapping vanishing cycles, rendering perturbation theory singular. In the parent theories, such singularities are of complex codimension one in moduli space, so one can go around them and continuously connect various phases. However, after orientifolding, these singularities become of real codimension 1, preventing different phases from being connected smoothly. Therefore, orientifolds constructed geometrically in the large radius regime do not necessarily all have orbifold CFT counterparts, and in particular may have different discrete properties. Indeed, we will see that this is the case for our model in §4. For this reason, and because we will stabilize all radii at finite distance away from the orbifold point anyway, we will first blow up the orbifold into a smooth, large Calabi-Yau threefold , using standard techniques in algebraic geometry. We then consider the string theory whose world-sheet definition is the sigma model with this Calabi-Yau target space, and then define an orientifold of this model. The result is similar to a toroidal orientifold but realizes discrete choices not possible in the CFT orientifold framework. Still, readers familiar with that framework may find it useful to read §4 to get an overview of the construction before proceeding. In the following, we will first study a local description valid near the orbifold fixed points. We give a completely explicit description of the resolution, the orientifold involution, and the brane embeddings. We also review how intersection numbers, important for example to derive the Kähler potential, are computed in this setup. We then move on to the compact Calabi-Yau and discuss its lift to F-theory on an elliptically fibered Calabi-Yau fourfold (which we define as M-theory in the limit of vanishing elliptic fiber area). Finally, we calculate the D3 tadpole for our model, and note an interesting geometrical transition where a 7-brane stack “eats” an O3-plane while changing its topology to preserve the net D3-brane charge. 2.1. Local model To understand the resolved geometry and the orientifold involution, it is useful to consider first a local model of the singularities, given by . The resolution of this orbifold can be described explicitly as a toric variety, following the general construction outlined in . The data underlying any three dimensional toric variety is given by a lattice and the choice of a fan , which is a collection of cones generated by lattice points in , satisfying the condition that every face of a cone is also a cone, and that the intersection of two cones is a face of each. The singular orbifold is described by the simple fan given in fig. 1, consisting of a single 3-dimensional cone generated by the lattice vectors , and . As usual for Calabi-Yau varieties, the third component of each vector equals 1, so we can restrict our attention to the other two coordinates, as we did in the figure. To each vertex a complex variable is assigned, and to each dimension a monomial . In this case, , , . The toric variety is then simply given by all not in a certain set , modulo complex rescalings that leave the invariant. The excluded set is given by the values of which have simultaneous zeros of coordinates not belonging to the same cone. Since there is only one three dimensional cone here, is empty. The only rescalings that leave the invariant are given precisely by (2.1). Thus, is indeed . The fact that is singular can be traced back to the fact that the top dimensional cone generators do not span the full lattice , since . To resolve the variety, one has to refine the fan such that all top dimensional cones have determinant 1. There are two distinct ways of doing this in the case at hand, one symmetric and one asymmetric, as shown in fig. 2 resp. . The dual graphs are shown in fig. 3. As we will review below, the vertices in fig. 2 can be thought of as divisors, the lines as curves at the intersections of two divisors, and the faces as points at the intersections of three divisors. In the dual graphs, the role of faces and vertices is interchanged. As shown in fig. 2 , there are now 6 vertices and 4 cones , , , all of determinant 1. The vertices are given by the matrix We associate complex variables to the and to the . The powers in the monomials are simply given by the rows of this matrix, i.e. , , . The rescalings leaving the invariant are with . The set of excluded points is again given by simultaneous zeros of coordinates not in the same cone, for example and are excluded, but is not. The toric variety is thus given by , with the actions given in (2.3). To each vertex corresponds a toric divisor, by setting the associated coordinate equal to zero. Curves are obtained by intersecting divisors, i.e. setting two coordinates to zero. To avoid being on the excluded locus , the corresponding vertices must be part of the same cone, in other words they have to be joined by a line in fig. 2. Compact curves correspond to internal lines. In the case at hand, there are three such curves, which we denote by , where and cyclic permutations thereof. Finally, triple intersection points of divisors are obtained by setting the 3 coordinates associated to a single cone to zero. Thus the triple intersection number is 1 for 3 distinct divisors belonging to the same cone, and 0 otherwise. The divisors are the original divisors we had in the unresolved variety, the are the exceptional divisors produced by the resolution, and the are the exceptional curves. The latter have topology . This can be seen as follows. for example is given by . To avoid the excluded set , we must take , and . This allows choosing a gauge with , so which is of course . Orientifold action and D-brane embedding Let us now look at the orientifold involution. There are several choices. We choose . The fixed points are then given by the for which for some . The following possibilities arise: (1) If all , we need and therefore . Because belong to the same cone, this is an allowed point. Thus, we get an isolated fixed point that can be represented by . This corresponds to an O3-plane. (2) If say , then to avoid the excluded set , we must take , , . Then (2.5) implies and , and imposes no further constraints. Therefore the entire divisor is fixed. This gives us an O7-plane. Similarly, there will be O7 planes on and . The topology of these divisors is easily determined. Since , and are all nonzero, we can fix the scaling gauge by setting these variables equal to 1. The divisor is then parametrized by the remaining variables and without further identifications, so it is a copy of . The action of on the exceptional s is also straightforward to determine. After a gauge transformation , the orientifold action can be written as , which acts on (2.4) as That is, the s are mapped to themselves in an orientation preserving way, with fixed points at the poles, where the intersects the O3 or O7 planes. Note that one cannot wrap a closed string once around a pole of the quotient , since the endpoints of a string can only be identified by an orientifold if the orientation of the string is reversed. Therefore the minimal closed string instanton wraps twice (or the original once). The instanton phase should furthermore be invariant under the orientifold action , which implies We still have to specify how we embed D-branes in this geometry. We will put D7-branes on top of the O7-planes such that D7 tadpole is canceled locally. We choose the O7-planes to be non-exotic, so each induces units of D7-brane charge in the Calabi-Yau (or in the quotient ), and we need a stack of 8 coincident D7-branes on each to cancel this. This gives rise to an gauge group on each divisor . Note that since the are disjoint in the resolved manifold, there are no massless bifundamentals from strings stretching between the D7-branes. To decide if there is massless adjoint matter, we need to know the topology of the in the compact geometry. We will get to this further on. To construct the Kähler potential on moduli space, we will need the triple intersection numbers of the divisors, including triple intersections involving identical divisors. These numbers also determine self-intersections of curves inside divisors, which characterize the local geometry. In this subsection we will review how to obtain these numbers. The reader who is only interested in the results can safely skip this part however. We can derive the intersection numbers in the local model for compact intersections. Linear combinations of divisors whose associated line bundle is trivial on the noncompact variety will give zero compact intersections with any combination of other divisors. Denoting the divisors collectively as , , and the corresponding coordinates by , we have that is a section of the line bundle . The invariant monomials are functions, so the corresponding is trivial. For the purpose of computing compact intersections, this implies three linear relations between our divisors†† Despite the abuse of notation, these relations between divisors should not be confused with the relations between the corresponding vertices (2.2)!: , and so on. We should emphasize that this relation does not mean that this linear combination of divisors will also be trivial in the compact geometry. Rather, it means that this linear combination does not intersect the compact curves, and hence can be moved away from the origin — in the compact geometry, such a divisor corresponds to a “sliding divisor” such as . These divisors descend directly from the unresolved , and are in this sense independent of the blowup. At any rate, these relations together with the triple intersection numbers of distinct divisors obtained directly from the fan are sufficient to determine all compact triple intersection numbers. This gives for example . From this, we also obtain the intersection numbers of the divisors with the compact curves defined above. These curves form a basis of the Mori cone i.e. the cone of effective holomorphic curves in . We get for their intersections: Note that the entries are precisely the charges of the rescalings (2.3). Indeed, the Mori cone intersection numbers always form a basis of the rescaling charges. This is an elementary algebraic consequence of the various definitions we made. The triple intersection numbers also give the self-intersection numbers of the curves inside the exceptional divisors, for example . Finally, apart from one subtlety, it is straightforward to deduce the intersection numbers of the orientifold quotient . The subtlety is the following. Denote the projection from to by . Naively one might think one should take the toric divisors of the quotient, considered as 2-forms, to be related to those of the double cover by . This is correct for the divisors , but not for the , for which we should take . This can be seen as follows. Because is fixed by the , its volume in must equal the volume of in the quotient. But the volume of is given by which is half the volume of . So we must take to correct for this. For the divisors on the other hand, whose volume does indeed get halved, there is no such correction factor of 2. Thus we get for example The half integral triple intersection product is possible because the intersection point coincides with the fixed point singularity (the O3). For the intersections of the Mori cone generators, we thus get The asymmetric resolution in fig. 2 can be treated in a completely analogous manner. The vertices of the fan remain the same, so the scalings (2.3) remain the same too. The cones themselves do change, so the excluded region will be different, as well as the intersection products. The generators of the Mori cone and their intersections are now given by†† To avoid cluttering, we drop the index ‘’ here. In section 3, where the relation between the two resolutions will be studied in more detail, the ‘+’ index will be reinstated. From this, we again get the self-intersections of the curves in the divisors: , , , . At the level of the intersections, the curves are related to those of the symmetric resolution by , , . These relations are characteristic of a flop; indeed, the symmetric and asymmetric resolutions are related by flopping the curve . The orientifold action is again . As in the symmetric resolution, the divisors support O7-planes. Now however there is no isolated fixed point: lies in the excluded set . All ’s are acted on by as in (2.6), except , which is pointwise fixed, since it is embedded in an O7-plane. The triple intersections of the quotient are obtained by the rules given earlier (i.e. add an overall factor of and ). This gives for the Mori cone 2.2. Compact model To get the compact model , one simply glues the 64 local models together, with transition functions determined by the transition functions between the -coordinates in the original . This gives exceptional divisors and O7-planes on divisors . Here (with ) and . On each O7-plane, we furthermore put an stack of D7-branes. This locally cancels the D7-tadpole, so the axio-dilaton is constant on . In the symmetric resolution, there are 64 O3-planes. In the asymmetric resolution, these are absent.†† By asymmetric resolution in the compact model, we mean the resolution obtained by blowing up each local patch in the same asymmetric way. In principle there could be mixed symmetric/asymmetric resolutions, but we will not consider these. The global topology of the various divisors is easily deduced. Let us consider for example the divisors in the symmetric resolution. The topology of the resolved manifold with the exceptional divisors removed is the same as the topology of with its singularities removed. In this space, the divisors have topology with the singularities removed, that is with four points removed in each factor. In each local patch, this looks like with the origin in each factor removed. From the explicit construction of the local model given above, it is clear that in the resolved space, the origin of each factor is simply put back as a point (as opposed to being replaced by some exceptional curve). Therefore, in the resolved compact model, the divisors are simply . For the topology of the exceptional divisors we get similarly blown up in 4 points (corresponding to the four intersections of a fixed line with the fixed planes in ). In the asymmetric resolution on the other hand the and divisors still have topology , but the divisors are now blown up in 16 points. The and divisors are , and is blown up in 8 points. All these divisors evidently have , since has this property, and blowing up only changes . This has important consequences: (1) There is no massless adjoint matter in the gauge theory. Since moreover the do not intersect, there is no massless bifundamental matter either. So the gauge theory is pure super Yang-Mills, and in particular will give rise to gaugino condensation and the generation of a nonperturbative superpotential for the Kähler moduli governing the size of the . (2) D3-instantons wrapping the exceptional divisors will have the minimal number of fermionic zero modes, and therefore contribute to the superpotential. To make this more precise, we need to consider the dual M-theory on a smooth Calabi-Yau fourfold, where the D3-instantons lift to M5-instantons. In this context it has been shown that if the M5 wraps a divisor satisfying (which in particular implies that its holomorphic Euler characteristic equals 1), there is a contribution to the superpotential . In section 3, we will prove in detail that this is indeed the case for the lifts of the D3-instantons wrapped on the exceptional divisors. We also give a short argument below. There are in fact other consistency conditions that need to be fulfilled. We will discuss these in section 6. 2.3. M/F-theory description of the model Type IIB string theory on the orientifold is dual to M-theory on an elliptically fibered Calabi-Yau fourfold with base , in the limit of vanishing fiber area. The dual fourfold is easily constructed in this case : it is simply , where the acts as our orientifold involution on , and as on . This gives a singular fourfold, with elliptic fibers degenerating to a singularity on top of the divisors , and, in the symmetric resolution, a degenerate fiber with four terminal singularities on top of each fixed point in . It can be considered as a partial resolution of . Again this is an example of a Borcea-Voisin model [21,100]. To rigorously address the question whether the lifts of the D3-instantons have the required properties mentioned in the previous subsection, one needs to resolve this fourfold in a way that preserves the elliptic fibration. This is somewhat tricky, and will be the subject of section 3. The basic idea is simple however. On the double cover of , the M5-brane lift of a D3 instanton wrapped on a divisor is . As argued in section 2.2, for and any of the divisors of interest discussed there. So the only harmonic -form on is the -form living on . Considering now the quotient in , we see that is odd and thus gets projected out. Moreover, blowing up the quotient singularities of will only change . Hence, also after resolving the fourfold, . Another important point in the arguments we will give for the nonvanishing of the instanton conributions is the fact that the M5-branes under considerations have trivial third cohomology. This can be argued similarly. On , the third cohomology is given by the product of and . But quotienting by projects out every such class because the elements of are even and those of are odd. Furthermore, blowing up will not add any new 3-cycles. So is trivial also after resolving the fourfold. A more precise discussion will be given in section 3. 2.4. D3 tadpole and O3-curvature transition We now compute the D3 tadpole measured in the quotient (as usual, in the double cover , is twice this). In the symmetric resolution, we have O3-planes. Choosing these to be non-exotic, their contribution to the D3-brane charge is The 7-branes also contribute to the D3 tadpole, through the “anomalous” couplings of RR-fields to worldvolume curvature [55,28,95,29,23,24]. In a Calabi-Yau threefold, a single D-brane wrapped around a divisor thus contributes a D3-charge , and an O7-plane . The total contribution from an O7 + D7 stack wrapped on is therefore in , and half of that in the quotient. Hence the total 7-brane contribution is In the symmetric resolution, has topology , so and There can also be contributions from the (half-integral quantized) -field to various tadpoles, as well as from gauge instantons on the D7-branes, but we will take here, in which case there are no tadpole contributions of this kind. Combining O3 and 7-brane contributions in this case gives In the asymmetric resolution and , so This agreement of tadpoles in symmetric and asymmetric resolutions can be understood locally: when flopping one local patch from fig. 3 to , one O disappears from the corresponding orientifold, so increases by in (2.12), but at the same time the Euler characteristic of the 7-branes wrapped around changes: a point gets blown up, which increases and therefore by 1, so decreases by in (2.13), and the total charge is conserved. If this transformation can be realized physically, this is a rather interesting phenomenon, in which a 7-brane stack “eats” an O3 and blows up to conserve the net D3-charge. This value of the tadpole fits nicely with the fourfold description. In this picture, the D3 tadpole is given by The Euler characteristic of the fourfold say for the asymmetric resolution can be computed for example as follows. Removing the divisors from the base together with the fibers on top gives a space which is a direct product with a factor. This has Euler characteristic 0. The Euler characteristic of the full space is therefore the sum of the Euler characteristics of the times the Euler characteristic of the fiber, which is 6 (it can be thought of as a collection of 5 spheres connected along 4 double points according to the extended Dynkin diagram). So and . A similar match can be made for the symmetric resolution after properly taking into account the contribution to from terminal singularities . The D3-tadpole thus produced can be canceled by adding 28 independent mobile D3-branes, or by turning on RR and NSNS 3-form fluxes. This is further discussed in section 5. In the symmetric resolution, we could also have chosen our O3-planes to be exotic. This does not change the geometry; it merely corresponds to turning on (torsion) twisted cohomology classes for the field strengths and/or in , where surrounds the O3 in [104,60,11]. In the M-theory dual this corresponds to turning on torsion -flux around the terminal -singularities . The D3-charge of any exotic O3 has the opposite sign of a normal O3. The total tadpole in this case is thus To cancel this, one needs 4 anti-D3 branes, which breaks supersymmetry. Incidentally, the O3s are required to be exotic for a consistent CFT description at the orbifold point, as we will discuss further in section 4. But, as stressed at the beginning of this section, it should not surprise us to find different consistent models at large radius. 3. Fourfold Geometry In this section we describe the geometry of the resolved Calabi-Yau fourfold. We describe the symmetric and asymmetric resolutions of the orbifold, and the birational transformation relating the two resolutions. To simplify the presentation, we start with a lower dimensional orbifold, , which is dual to IIB on an orientifold of . We discuss the resolution of the orbifold and the properties of the exceptional divisors introduced in the blow-up process. We then move to our main example, the orbifold. We present two distinct resolutions and discuss their elliptic fibration structure. Starting from local models, we discuss the birational factorizations of the transformations relating the elliptically fibered Calabi-Yau fourfolds and their bases. This somewhat technical analysis is necessary to prove that the exceptional divisors in the (singular) symmetric resolution have the right topological properties to contribute to the nonperturbative superpotential: they have holomorphic Euler characteristic and the higher cohomology groups , vanish. We also show that their third cohomology is trivial, which is important for arguing that the instanton prefactor is nonvanishing. 3.1. Lower dimensional orientifold It is instructive to consider first the lower dimensional analog, namely F-theory on . The action of the orbifold group is presented in (2.1). We can view as an elliptic fibration over : let and be the coordinates on the base and be the coordinate on the elliptic fiber. Then, the elliptic fiber degenerates to type fibers†† That is, along the fixed point set, the fiber degenerates to , which is a rational curve with four singular points. along the fixed locus of and in the base. In F-theory, such a singularity corresponds to an gauge group [26,15]. The base is and there are lines of fibers intersecting at points†† The corresponding Weierstrass model describing the transverse collision of two fibers is not minimal.. In order to obtain a smooth Calabi-Yau threefold, we need first to blow-up the base at these points and then resolve the singularities of the elliptic fibration. Let us first discuss the blowing-up of the base . We can work in local coordinates in the fibration around the point , which lies at the intersection of the fixed lines and . To describe the blow-up, introduce two coordinate patches and as follows: The coordinates and are homogeneous coordinates on the exceptional . The actions lift to the blown-up threefold; there are two fixed points on the exceptional given by and , where it intersects the unresolved singular divisors. The elliptic fiber over the exceptional is smooth, except at the points and , where there are singularities. We note that the elliptic fibration over the blown-up base admits a section. The blow-up process is ilustrated in the figure below. Blowing-up the base introduce new Kähler parameters. The next step is to resolve the elliptic fibration and this will introduce additional Kähler parameters since after the blow-ups in the base there are isolated curves on top of which the elliptic fiber is type . Taking into account the original 2 Kähler parameters of the base and the section, we recover the Kähler parameters of the resolution. We have obtained a smooth threefold that is elliptically fibered and is one of the Borcea-Voisin models [21,100]. The process of resolving the elliptic fiber will turn the elliptic fibration over the exceptional divisors in the base into rational elliptic surfaces , , that is del Pezzo surfaces . These have . In type theory language, the description of the orientifold of is as follows . The fixed curves are wrapped by D-branes sitting on top of orientifold O planes, and on the worldvolume of each -brane there is an gauge theory. After blowing-up the base, the gauge theory is with no matter. Proceeding analogously to the previous section, consider now the orbifold . The orbifold group acts as This is another example of the Borcea-Voisin construction [21,100]. To get our Calabi-Yau fourfold start with the Calabi-Yau orbifold with the orbifold action given by (3.1) and construct , where is an involution of that changes the sign of the holomorphic three-form. The local singularities are of the form . The figure below presents the toric resolutions of the singularities. We see again that the local singularities do admit crepant resolutions and it is possible to glue them together and get a smooth crepant resolution with and . We can also think of as an elliptic fibration over , with fibers along the fixed point set of , and .

View Answer Discuss. Use inductive reasoning to make a conjecture about the sum of a number and itself. Inductive and Deductive Reasoning Inductive Reasoning Inductive reasoning is one method of reasoning that researchers use. Monthly Downloads for the past 3 years . * Mrs Jennifer's house is somewhere to the left of the green marbles one and the third one along is white marbles. step 3 is wrong Posted in LOGIC TRICK EQUATION #2 - Hard Logic Chess Puzzle Assume you have the white pieces, can you win in a half a move ? Inductive reasoning means coming to a very broad conclusion based on just a few observations. (ii) Write q -> p in words. Conversely, deductive reasoning uses available information, facts or premises to arrive at a conclusion. You may want to discuss the links among reasoning, evidence, and proof at that point. In math, you can support your proof with supporting proofs or … You know it'll be true. Deductive reasoning goes from a general to a specific instance. Problem 3 : Let p be "the value of x is -5" and let q be "the absolute value of x is 5". Noisy Deductive Reasoning: How Humans Construct Math, and How Math Constructs Universes. 2 3. Inductive Reasoning. Deductive Reasoning Logical Problem. In Math in Action on page 15 of the Student Book, students will have an opportunity to revisit an investigative scenario through conjectures, witness statements, and a diagram. Reply. Therefore, the test will be easy. Thus, the premises used in deductive reasoning are in many ways the most important part of the entire process of deductive reasoning, as was proved by the help of the above given examples. You are given a triangle to work with. 10. Then, from that rule, we make a true conclusion about something specific. deductive reasoning, inductive reasoning, valid argument, logical argument, conjecture, verify, proof, prove, disprove, counterexample, observation, undefined term, postulate, theorem (G.1) Student/Teacher Actions (what students and teachers should be doing to facilitate learning) 1. Being able to use deductive reasoning is valuable to employers. Inductive Reasoning: My mother is Irish. Deductive and Inductive Reasoning Asked by a student at Winona Senior High School on January 28, 1998: I was talking with my geometry teacher the other day and we discussed inductive and deductive reasoning. Deductive reasoning is the process by which a person makes conclusions based on previously known facts. In science, you can then support your conclusions with experimental data. Deductive Reasoning – Drawing a specific conclusion through logical reasoning by starting with general assumptions that are known to be valid. Review the basic vocabulary included on the sheets. Deductive Reasoning Deductive reasoning is the process of reasoning logically from given statements to make a conclusion. Deductive reasoning requires one to start with a few general ideas, called premises, and apply them to a specific situation. February 2, 2016 February 2, 2016 Todd Abel explicit rules, inductive reasoning, math teaching, pattern-sniffing, recursive rules, standards of mathematical practice Leave a comment One of the principle algebraic ways of thinking that we came up with during the introductory problems was pattern-sniffing . How is it used in Mathermatics? The comparatively poor performance of American students on international math exams means the country should spend more money on math education. If you get an A on your math test, then you can go to the movies. INDUCTIVE AND DEDUCTIVE REASONING WORKSHEET. Therefore, Mr. D is over 7 feet tall. Deductive - Displaying top 8 worksheets found for this concept.. (deposited 26 Nov 2020 05:22) [Currently Displayed] Monthly Views for the past 3 years. Deductive reasoning, also deductive logic, is the process of reasoning from one or more statements (premises) to reach a logical conclusion.. Deductive reasoning goes in the same direction as that of the conditionals, and links premises with conclusions.If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true. There are 4 big houses in my home town. All math teachers are over 7 feet tall. Therefore, the second lipstick I pull from my bag will be red, too. Thus, if they are wrong, the entire foundation of the whole line of reasoning is faulty and thus, the conclusions derived will also be faulty. Thus, it produces from the specific to the general. Recognized rules, laws, theories, and other widely accepted truths are used to prove that a conclusion is right. When you reason deductively, you can say “therefore” with certainty. Pretty hard to see a pattern when pieces are missing. Law of detachment : If p -> q is a true conditional statement and p is true, then q is true. The argument is valid, but is certainly not true. Syllogisms are a form of deductive reasoning that help people discover a truth. Can you help me answer this question? A logical inference is a connection from a first statement (a “premise”) to a second statement (“the conclusion”) for which the rules of logic show that if the first statement is true, the second statement should be true. Therefore, all the lipsticks in my bag are red. You then conclude that every goose is white. Plum Analytics. When math teachers discuss deductive reasoning, they usually talk about syllogisms. Deductive reasoning is the type of reasoning used when making a Geometric proof, when attorneys present a case, or any time you try and convince someone using facts and arguments. Law of syllogism : If p -> q and q -> p are true conditional statements, p->q is true. In each question you will be presented with a logical sequence of five figures. Employers value decisive, proactive employees. The above examples are of the form If p, then q. You will have 25 minutesin which to correctly answer as many as you can. Deductive reasoning uses facts, definitions, and accepted properties in a logical order to write a logical statement. Predict the next number. Deductive Reasoning. In deductive reasoning, the conclusions are certain, whereas, in Inductive reasoning, the conclusions are probabilistic. Inductive Reasoning. Deductive reasoning is introduced in math classes to help students understand equations and create proofs. In the Inductive method of mathematical reasoning, the validity of the statement is checked by a certain set of rules and then it is generalized. Examples of Inductive Reasoning Some examples Every quiz has been easy. B and C are the same but C is correct? Browse inductive reasoning math resources on Teachers Pay Teachers, a marketplace trusted by millions of teachers for original educational resources. Still, they are often juxtaposed due to lack of adequate information. Deductive Reasoning 3. If you go to the movies, then you can watch your favorite actor. Inductive reasoning - Think of it like a We start with specifics and move to generalities Deductive reasoning – think of it like a We start with generalities and move to specifics. Mr. D. is a math teacher. The principle of mathematical induction uses the concept of inductive reasoning. It is, in fact, the way in which geometric proofs are written. Two Laws of Deductive Reasoning (i) Law of detachment (ii) Law of syllogism. You can use deductive reasoning in a science class or a math class to test an existing theory or hypothesis. Inductive Reasoning Free Sample Test 1 Solutions Booklet AssessmentDay Practice Aptitude Tests Difficulty Rating: Difficult . The sum of any triangle’s three angles is 180 degrees. Problem 1 : Sketch the next figure in the pattern. Benefits of Deductive Reasoning . If … He wanted me to find out exactly what they are and find an example just to see if I could do it. Even when the decision doesn't work out, you can explain why you decided to do what you did. Actions (login required) View Item: … Every row and column contain the same figures/numbers. When we deduce something, we take a rule and apply it to a unique situation. Deductive Reasoning Puzzles With Answers #1 - Tricky Math Problem 1 dollar = 100 cent = 10 cent x 10 cent = 1/10 dollar x 1/10 dollar = 1/100 dollar = 1 cent => 1 dollar = 1 cent solve this tricky problem ? Instructions. Inductive reasoning is the opposite of deductive reasoning. Inductive Reasoning: The first lipstick I pulled from my bag is red. Deductive Reasoning The process of reasoning from known facts to conclusions. Inductive vs. Deductive Reasoning 1 2. Use your logical reasoning skills to fill the missing cells of the latin square. Foundations in Math 110 Section 1.4 Proving Conjectures: Deductive Reasoning Proof – A mathematical argument showing that a statement is valid in all cases, or that no counterexample exists. All research that makes inference or generalizations about the results of a study uses inductive reasoning (Berg & Latin, 2008). Deductive reasoning moves from generalized statement to a valid conclusion, whereas Inductive reasoning moves from specific observation to a generalization. Referring to the practice inductive reasoning – question three with the Hershy kiss things. (major premise) x is p. (minor premise) Therefore, x is q. Clear examples and definition of Deductive Reasoning. Explanation. The second lipstick I pulled from my bag is red. So, in maths, deductive reasoning is considered to be more important than inductive. Deductive reasoning is often used to make inferences in science and math, as you must use formal logic to support a conclusion or a solution. Deductive reasoning allows you to use logic to justify work-related decisions. Deductive reasoning uses facts, defi nitions, accepted properties, and the laws of logic to form a logical argument. It is based on making a conclusion or generalization based on a limited number of observations. What does Conjecture mean? These two logics are exactly opposite to each other. Have you heard of Inductive and Deductive Reasoning? In this article, we are going to tell you the basic differences between inductive and deductive reasoning, which will help you to understand them better. Inductive Reasoning. Test your IQ with this deductive reasoning test using latin squares. Problem 2 : Describe a pattern in the sequence of numbers. A latin square has two important properties: A row or column never contains the same figure/number twice. Here’s an example. The concept of deductive reasoning is often expressed visually using a funnel that narrows a general idea into a specific conclusion. That’s why we call deduction top-down logic—you move from the general to the particular. Inductive reasoning is typified by the following example: Suppose every goose you observe throughout your lifetime is white. Note that this conclusion is not 100% definite. This inductive reasoning test comprises 22 questions. View Answer Discuss. All lipsticks in my bag are red. Deductive Reasoning: The first lipstick I pulled from my bag is red. (i) Write p -> q in words. This is an inductive conclusion. They are made from these materials: red marbles, green marbles, white marbles and blue marbles. Deductive reasoning starts with some general observations and deducts (wipes away) every unnecessary distraction to leave a specific, valid conclusion. Distribute copies of the two activity sheets. Making assumptions. their conjectures through the use of deductive reasoning. Forget reasoning – proof read. Deductive reasoning, or deduction, is one of the two basic types of logical inference. Also, on question 2 (same test) with square rotating clockwise three and ball counter clockwise two – there is no ball in picture two. You're starting with facts, and then you're deducing other facts from those facts. Induction, by contrast, is bottom-up logic. Inductive vs deductive reasoning 1. Inductive reasoning is a very different beast. Deductive Reasoning Startswith a general rule (a premise) which we know to be true. For example, if we say all primes other than two are odd, deductive reasoning would let us say that 210000212343848212 is not prime. The teacher used PowerPoint in the last few classes. Deductive reasoning, unlike inductive reasoning, is a valid form of proof. Paneer Capsicum Samosa Recipe, Russet Potatoes Substitute Nz, Lg Lsg4511st Parts, What Do Holly Berries Taste Like, Chemicals Between Us Soundtrack, Mjolnir Necklace Australia, Famous Effective Speakers, Tribal Hammerhead Shark Tattoo Meaning,

If the applied voltage is slightly larger than the LEDâs forward voltage, the forward voltage exceeds the recommended value, which can be 1.5 to 4 volts for LEDs of different colors. This calculator works out the resistor value to accompany an LED by entering the battery voltage along with the LED forward voltage and typical current. In this case, enter any two of the following values: the voltage across the resistor, the current through the resistor, or its resistance in ohms to find the power dissipation in watts. Resistor precision: select the desired standard resistor precision: 10% (E12), 5% (E24), 2% (E48) or 1% (E96). LED Calculator. To calculate the power rating of the resistor we have to use Jouleâs power law: Based on the calculation, then 1/8W resistor will work just fine, though a 1/4W resistor may be easier to get your hands on. LED Calculator 3D MHS Builder Gift Certificates TCSS Theme Song. It will calculate the power dissipated by the resistor and LED(s), the recommended resistor Wattage, the total power consumed by the circuit and the efficiency of the design (Power consumed by the LED(s) / Total circuit power consumption) x 100). Overdriving, even briefly, will significantly reduce it's life and light output. When calculating the required resistance of the current limiting resistor Rs, all voltage drops across each LED needs to be considered. Connecting an LED directly to the power supply will burn it out in a heartbeat. However, we do not guarantee that our converters and calculators are free of errors. There are several ways to identify the leads of an LED: How to test diodes, transistors, Zeners, LEDs and MosFets, The cathode (negative) is usually marked with a. The Unit Conversion page provides a solution for engineers, translators, and for anyone whose activities require working with quantities measured in different units. This LED resistor calculator calculates the value of the resistor which you would need in order to produce the desired current to go through the LED. All the calculators in step 2 are just doing some simple math that you can do at home: The formula to calculate resistance in a circuit is: R=V/I or, more relevant to what we're doing: (Source Volts - LED Volts) / (Current / 1000) = Resistance * So if we have a 12v battery powering a 3.5V 25mA LED our formula becomes: (12 - 3.5) / (25 / 1000) = 340ohms. number of LEDs: The resistance of the current limiting resistor for strings with fewer LEDs than the max. Use our resistor color code calculator to find out the color bands for different (20%, 0.5%...) precision resistors. TranslatorsCafe.com Unit Converter YouTube channel, Terms and Conditions They are the nearest (upper and lower) standard values closest to the raw calculated resistance. LED Resistor Calculator is the perfect solution for you, bundled with a huge feature set, yet retaining simplicity and ease of use. Current Limiting Resistor Calculator for LEDs For Single and Multiple LED Circuits. A light-emitting diode (LED) is a semiconductor light source with two or more leads. This LED resistor calculator will help you to pick up a right value of resistor for the LED in your LED circuit, you just have to input the values of Source voltage (V s), LED forward current (I f) and Led forward voltage (V f). The value of the series current limiting resistor Rs can be calculated using Ohmâs law formula in which the supply voltage Vs is offset by the forward voltage drop across the diode Vf: where Vs is the power supply voltage (for example 5 V USB power) in volts, Vf is the LED forward voltage drop in volts and I is the LED current in amperes. To get started, input the required fields below and ⦠This LED calculator will help you design your LED array and choose the best current limiting resistors values. For example, if a voltage drop across each illuminated LED is 2 V and we connected five LEDs in series, then the total voltage drop across all five will be 5 à 2 = 10 V. Several identical LEDs can be also connected in parallel. All of the content is provided âas isâ, without warranty of any kind. This is the minimum power rating you can use on your resistor. LED Resistor Calculator. We work hard to ensure that the results presented by TranslatorsCafe.com converters and calculators are correct. You have to use only one in your circuit - it's best to select the one which is closer (the one with * after the value). 2V for a standard red LED; 3.6V for a white LED used in lighting, stroboscope, etc. LEDs and resistors behave very differently in circuits. The ballast resistor is used to limit the current through the LED and to prevent that it burns. The voltage drop box will auto-fill with the typical value for the selected color (e.g. All LEDs require some form of current limiting. Calculation of Current-Limiting Resistors for a Single LED and LED Arrays This LED calculator designs a simple one-LED with one series resistor or multi-LED array circuit with series-LED chains combined in parallel clusters. KR Pixel Stick V2 34.75" Price: $84.99. Current Limiting Resistor Calculator for Leds. At present (2018), it can be observed that LED street lights, with a planned service life of 10 years, serve no more than a year. LED Resistor Calculator. The power supply voltage must be higher than the LED forward voltage and lower than 250 V. Power LEDs require constant current driver instead of series current limiting resistors. Usually, the resistor wattage is selected close to twice the value calculated here. Superbright leds can go from 30mA up to several amps. A current limiting resistor regulates and reduces the current in a circuit. For multiple LEDs a second drop-down will appear where you can select either a series or parallel connection. Supply voltage: Type in a voltage greater than the LED voltage drop for a single LED circuit and parallel connection or the sum of all voltage drops when connecting multiple LEDs in series. In this common LED street lighting fixture 8 strings of 5 powerful LEDs for a total of 40 LEDs are driven by an efficient constant current power supply; note that two strings (top left and bottom right) are dark in this fixture installed only a couple of months ago because in each of them one diode failed and protection devices are not used or not working, 3014 (3.0 à 1.4 mm) SMD LED used in LCD TV with LED backlight, Flexible LED displays in a public place; a LED display uses an array of light-emitting diodes as pixels; because of LED very high brightness, they are commonly used outdoors as billboards or highway destination sights that are visible in bright sunlight. © ANVICA Software Development 2002â2020. The current through each diode is identical, which ensures uniform brightness. If the voltage source is equal to ⦠If you just began with electronics like some DIY stuff or Arduino, then probably the first project or circuit you might have built would be to blink an LED. The resistance of the current limiting resistor for strings with the max. If you want to connect the LEDs in parallel each one should have its own resistor. Typical current of LEDs used for indication is 20 mA. Connecting an LED directly to the power supply will burn it out in a heartbeat. A resistor of either power rating will work. If the voltage reaches the characteristic forward voltage value shown in the specifications, the LED âturns onâ and its resistance quickly drops off. The good thing about Arduino is that it has an on-board LED connected to Digital IO Pin 13 and all you need to do is to just plug-in the Arduino UNO board to a computer and upload the Blink Sketch. If the voltage across the resistor is increasing, the current is also proportionally increasing (we assume that the resistor value stays the same). Source is equal to ⦠calculate the minimum and maximum values based on the packaging resistor some... 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Recent studies have extended the theory of affine processes to the stochastic Volterra equati- ons framework. In this talk, I will describe how the theory of polynomial processes extends to the Volterra setting. In particular, I will explain the moment formula and an interesting stochastic invariance result in this context. This is joint work with Eduardo Abi Jaber, Christa Cuchiero, Luca Pelizzari and Sara Svaluto-Ferro. Polynomial Volterra processes Geometric properties of some rough curves via dynamical systems: SBR measure, local time and Rademacher chaos We investigate geometric properties of graphs of Takagi type functions, repre- sented by series based on smooth functions. They are Hölder continuous, and can be embedded into smooth dynamical systems, where their graphs emerge as pull- back attractors. It turns out that occupation measures and Sinai-Bowen-Ruelle (SBR) measures on their stable manifolds are dual by ’time’ reversal. A suitable version of approximate self-similarity for deterministic functions al- lows us to ’telescope’ small-scale properties from macroscopic ones. As one conse- quence, absolute continuity of the SBR measure is seen to be dual to the existence of local time. The investigation of questions of smoothness both for SBR as for oc- cupation measures surprisingly leads us to the Rademacher version of Malliavin’s calculus, Bernoulli convolutions, and into probabilistic number theory. The link between the rough curves considered and smooth dynamical systems can be gen- eralized in various ways. For instance, Gaussian randomizations of Takagi curves just reproduce the trajectories of Brownian motion. Applications to regularization of singular ODE by rough signals are on our agenda. Optimal consumption with labor income and borrowing constraints for recursive preferences In this talk, we present an optimal consumption and investment problem for an investor with liquidity constraints who has isoelastic recursive Epstein-Zin utility preferences and re- ceives a stochastic stream of income. We characterize the optimal consumption strategy as well as the terminal wealth for recursive utility under dynamic liquidity constraints, which pre- vent the investor to borrow against his stochastic future income. Using duality and backward SDE methods in a possibly non-Markovian diffusion model for the financial market, this gives rise to an interplay of singular control and optimal stopping problems. Our analysis extends to more general liquidity constraints. (Joint work with Dirk Becherer and Olivier Menoukeu Pamen) On two Formulations of McKean–Vlasov Control with Killing We study a McKean–Vlasov control problem with killing and common noise. The particles in this control model live on the real line and are killed at a positive intensity whenever they are in the negative half-line. Accordingly, the interaction between particles occurs through the subprobability distribution of the living particles. We establish the existence of an optimal semiclosed-loop control that only depends on the particles’ location and not their cumulative intensity. This problem cannot be addressed through classical mimicking arguments, because the particles’ subprobability distribution cannot be reconstructed from their location alone. Instead, we represent optimal controls in terms of the solutions to semilinear BSPDEs and show those solutions do not depend on the intensity variable. Generalized Front Propagation for Stochastic Spatial Models Nonlinear Diffusions and their Feller Properties Motivated by Knightian uncertainty, S. Peng introduced his celebrated G–Brownian motion. Intuitively speaking, it corresponds to a dynamic worst case expectation in a model where volatility is uncertain but postulated to take values in a bounded interval. Natural extensions of the G–Brownian motion are nonlinear diffusions, whose volatility (and drift) takes values in a random set that is allowed to depend on the canonical process in a Markovian way. Nonlinear diffusions satisfy the dynamic programming principle, which entails the semigroup property of a corresponding family of sublinear operators. In this talk, we discuss regularity properties of these semigroups that allow us to relate them to evolution equations. In particular, we explain a novel type of smoothing property and a stochastic representation result for general sublinear semigroups with pointwise generators of Hamilton-Jacobi-Bellman type. Latter also implies a unique characterization theorem for such semigroups. The talk is based on joint work with Lars Niemann (University of Freiburg). Numeraire-invariance and the law of one price in mean-variance portfolio selection and quadratic hedging In classical optimal transport, the contributions of Benamou-Brenier and Mc- Cann regarding the time-dependent version of the problem are cornerstones of the field and form the basis for a variety of applications in other mathematical areas. Stretched Brownian motion provides an analogue for the martingale version of this problem. We provide a characterization in terms of gradients of convex functions, similar to the characterization of optimizers in the classical transport problem for quadratic distance cost. Based on joint work with Julio Backhoff-Veraguas, Walter Schachermayer and Bertram Tschiderer.

Fixed Effects versus Lagged Dependent Variables Fixed effects and differences-in-differences estimators are based on the presumption of time-invariant (or group-invariant) omitted variables. Suppose, for example, we are interested in the effects of participation in a subsidized training program, as in the Dehejia and Wahba (1999) and Lalonde (1986) studies discussed in section (3.3.3). The key identifying assumption motivating fixed effects estimation in this case is E(y0it&i; Xit, Dit) — E(Y0it&i; Xit), (5.3.1) where a. i is an unobserved personal characteristic that determines, along with covariates, Xit, whether individual i gets training. To be concrete, & might be a measure of vocational skills, though a strike against the fixed-effects setup is the fact that the exact nature of the unobserved variables typically remains somewhat mysterious. In any case, coupled with a linear model for E(Yoit&i, Xit), assumption (5.3.1) leads to simple estimation strategies involving differences or deviations from means. For many causal questions, the notion that the most important omitted variables are time-invariant doesn’t seem plausible. The evaluation of training programs is a case in point. It seems likely that people looking to improve their labor market options by participating in a government-sponsored training program have suffered some kind of setback. Many training programs explicitly target people who have suffered a recent setback, e. g., men who recently lost their jobs. Consistent with this, Ashenfelter (1978) and Ashenfelter and Card (1985) find that training participants typically have earnings histories that exhibit a pre-program dip. Past earnings is a time-varying confounder that cannot be subsumed in a time-invariant variable like &i. The distinctive earnings histories of trainees motivates an estimation strategy that controls for past earnings directly and dispenses with the fixed effects. To be precise, instead of (5.3.1), we might base causal inference on the conditional independence assumption, E(y0itYit— Xit; Dit) — E(y0it Yit— h; Xit). (5.3.2) This is like saying that what makes trainees special is their earnings h periods ago. We can then use panel data to estimate where the causal effect of training is ft. To make this more general, Yu-h can be a vector including lagged earnings for multiple periods.9 Applied researchers using panel data are often faced with the challenge of choosing between fixed-effects and lagged-dependent variables models, i. e., between causal inferences based on (5.3.1) and (5.3.2). One solution to this dilemma is to work with a model that includes both lagged dependent variables and unobserved individual effects. In other words, identification might be based on a weaker conditional independence assumption: E(Y0itai; Yit—h; Xit; Dit) = E(Y0it^i; Yit—h; Xit), (5.3.4) which requires conditioning on both a. i and Yit-h. We can then try to estimate causal effects using a specification like Unfortunately, the conditions for consistent estimation of ft in equation (5.3.5) are much more demanding than those required with fixed effects or lagged dependent variables alone. This can be seen in a simple example where the lagged dependent variable is Yit_ 1. We kill the fixed effect by differencing, which produces The problem here is that the differenced residual, AVit, is necessarily correlated with the lagged dependent variable, AYit_i, because both are a function of Vit_1. Consequently, OLS estimates of (5.3.6) are not consistent for the parameters in (5.3.5), a problem first noted by Nickell (1981). This problem can be solved, though the solution requires strong assumptions. The easiest solution is to use Yit_2 as an instrument for Ayit_ 1 in (5.3.6). But this requires that Yit_2 be uncorrelated with the differenced residuals, AVit. This seems unlikely since residuals are the part of earnings left over after accounting for covariates. Most people’s earnings are highly correlated from one year to the next, so that past earnings are an excellent predictor of future earnings and earnings growth. If Vit is serially correlated, there may be no consistent estimator for (5.3.6). (Note also that the IV strategy using Yit_2 as an instrument requires at least three periods to obtain data for t, t — 1, and t — 2). Given the difficulties that arise when trying to estimate (5.3.6), we might ask whether the distinction between fixed effects and lagged dependent variables matters. The answer, unfortunately, is yes. The fixed-effects and lagged dependent variables models are not nested, which means we cannot hope to estimate one and get the other as a special case if need be. Only the more general and harder-to-identify model, (5.3.5), nests both fixed effects and lagged dependent variables.. So what’s an applied guy to do? One answer, as always, is to check the robustness of your findings using alternative identifying assumptions. That means that you would like to find broadly similar results using both models. Fixed effects and lagged dependent variables estimates also have a useful bracketing property. The appendix to this chapter shows that if (5.3.2) is correct, but you mistakenly use fixed effects, estimates of a positive treatment effect will tend to be too big. On the other hand, if (5.3.1) is correct and you mistakenly estimate an equation with lagged outcomes like (5.3.3), estimates of a positive treatment effect will tend to be too small. You can therefore think of fixed effects and lagged dependent variables as bounding the causal effect you are after. Guryan (2004) illustrates this sort of reasoning in a study estimating the effects of court-ordered busing on Black high school graduation rates.

Note: This is updated from earlier in the day. See below for Dolphins notes from Tuesday afternoon. # # # Dumbest remark of the playoffs so far? Had to be Thunder forward Serge Ibaka saying that LeBron James "is not a good defender." That's the same LeBron James who earned the most votes in balloting for the NBA's all-defensive team. James responded Tuesday morning. "I don't really care what he says," LeBron said. "He's stupid. Everyone says something to me every series. Then [the media] keeps trying to get a quote. I don't care what he says. It's stupid." Ibaka, per the Post, went on to the say that James "can play defense for two to three minutes but not 48.... LeBron can't play [Kevin Durant]" one-on-one. Responded James: "First of all, I'm not playing 48 minutes and K.D.'s not playing 48 minutes. I'm not sitting there saying I'm a Durant stopper because there's no such thing." Durant missed four of five shots when James was guarding him in the fourth quarter of Game 3. Durant was asked Monday what James did to him in the fourth quarter. "Nothing," he said. And the news conference ended thusly. It will be interesting to see how Ibaka explains himself. Some backtracking is likely, or at least a scolding by Scott Brooks. DOLPHINS PRACTICE NOTES ### Chad OchoCinco made three receptions at the first day of the Dolphins' three-day minicamp on Tuesday, including two terrific catches on the sideline, both from David Garrard. He jumped over Vontae Davis to make one of them. "He's fast, he's quick, and he's got that attacking mentality you love in a wide receiver," Matt Moore said. ### He spoke to reporters less than two minutes afterward. He said he wants to get "back to who we're all used to seeing and how I became what I am. I think I kind of lost that. For me, it's about getting back to basics, getting back to where it started. I'm looking to go back to Chad Johnson and just make it live again." ### OchoCinco said playing for the Dolphins "has been a childhood dream of mine growing up watching the Marks Brothers, watching Dan Marino. To be able to wear teal and orange, it's a pretty good feeling." ### He cracked, "I'm developing Brokeback Mountain chemistry with the players." He ended his session with reporters by saying: "I love you. Enjoy the show." ### Asked how OchoCinco is learning the playbook - which was a problem for him in New England - Joe Philbin said, "So far, so good." Philbin added: "It looks like he's fitting in well." ### David Garrard took the majority of the first team snaps but said he and Matt Moore are splitting them about evenly overall, with Ryan Tannehill receiving very few with the starters. Garrard looked the best of the three on Tuesday, which was also the case during the last practice open to the media, eight days ago. ### Garrard led a touchdown drive that ended with a rollout and strong throw to Roberto Wallace for a score. The offense was forced to punt on Matt Moore's final drive. ### Garrard's only glaring error was an interception to Sean Smith, who would have returned it for a touchdown if the play had been allowed to continue. Smith has been very impressive all offseason. "I like what I've seen from Sean," Philbin said. "His approach has been serious." ### Matt Moore also threw what would have been an interception return for a touchdown, to Karlos Dansby. ### Tannehill made some sharp passes - to tight end Les Brown, Marlon Moore, Wallace and others - but also threw a pick to Jason Trusnik. ### Jonathan Wade, competing for Nolan Carroll for the No. 4 cornerback job, made three terrific plays to break up potential receptions. And rookie Josh Kaddu, who couldn't join the team until this month because of his Oregon class schedule, looked sharp - including a would-be sack of Tannehill. (We say would-be because sacks aren't allowed.) Kaddu also knocked away a Tannehill pass over the middle. ### Wallace made several impressive catches, and Garrard raved about him afterward. Also, Philbin spoke very highly of receiver Legedu Naanee, noting "you like his size and ability and he can attack the middle of the field." ### Jonathan Martin, the second-rounder out of Stanford, got a lot of work with the first team at right tackle, with Artis Hicks at right guard. Lydon Murtha also got some first-team work at right tackle. And Martin got some snaps in relief of Jake Long at left tackle, which was Martin's position at Stanford. ### Reshad Jones and Chris Clemons remain the first-team safeties, backed up by Jimmy Wilson and Tyrone Culver. Free-agent pickup Tyrell Johnson continues to work with the third team. ### Rookie third-round tight end Michael Egnew, who was used a lot as a blocker in earlier practices, made two receptions across the middle, showing nice burst after the catch. ### Dan Carpenter kicked a 59-yard field goal but missed, to the left, from 57. COLLEGE FOOTBALL ITEM ### Encouraging news for the Orange Bowl: College presidents are expected to approve a four-team playoff, with plans to be announced perhaps as early as this week, and ESPN reported Monday night that the decision-makers are leaning toward incorporating the semifinals within the current bowl system, meaning the Orange, Fiesta, Rose and Sugar would alternate as hosts. It's not definite by any means, but the OB certainly would take that scenario. The national championship game is expected to be put up for bid, with Cowboy Stadium among the potential suitors. The new format would take effect with the 2014 season.

Numerical analysis of distributed optimal control problems governed by elliptic variational inequalities A continuous optimal control problem governed by an elliptic variational inequality was considered in Boukrouche-Tarzia, Comput. Optim. Appl., 53 (2012), 375-392 where the control variable is the internal energy . It was proved the existence and uniqueness of the optimal control and its associated state system. The objective of this work is to make the numerical analysis of the above optimal control problem, through the finite element method with Lagrange’s triangles of type 1. We discretize the elliptic variational inequality which define the state system and the corresponding cost functional, and we prove that there exists a discrete optimal control and its associated discrete state system for each positive (the parameter of the finite element method approximation). Finally, we show that the discrete optimal control and its associated state system converge to the continuous optimal control and its associated state system when the parameter goes to zero. Key words: Elliptic variational inequalities, distributed optimal control problems, numerical analysis, convergence of the optimal controls, free boundary problems. 2010 AMS Subject Classification 35J86, 35R35, 49J20, 49J40, 49M25, 65K15, 65N30. We consider a bounded domain whose regular boundary consists of the union of two disjoint portions and with meas ( ) . We consider the following free boundary problem : where the function in (1.1) can be considered as the internal energy in , is the constant temperature on and is the heat flux on . The variational formulation of the above problem is given as: Find such that We note that is a bilinear, continuous, symmetric on and a coercive form on , that is to say: there exists a constant such that In , the following continuous distributed optimal control problem associated with or the elliptic variational inequality was considered: Problem : Find the continuous distributed optimal control such that where the quadratic cost functional is defined by: with a given constant and is the corresponding solution of the elliptic variational inequality (1.3) associated to the control . Several continuous optimal control problems are governed by elliptic variational inequalities, for example: the process of biological waste-water treatment; reorientation of a satellite by propellers; and economics: the problem of consumer regulation of a monopoly, etc. There exist an abundant literature for optimal control problems [4, 42, 50], for optimal control problems governed by elliptic variational equalities or inequalities [2, 3, 5, 6, 7, 8, 9, 11, 19, 20, 26, 28, 30, 32, 34, 38, 40, 45, 46, 52, 53, 54], for numerical analysis of variational inequalities or optimal control problems [10, 13, 14, 15, 16, 17, 21, 22, 23, 24, 25, 27, 33, 35, 36, 37, 43, 47, 48, 49, 51], and for the numerical analysis of optimal control problems governed by an elliptic variational inequality there exist a few numbers of papers [1, 29, 31, 44]. The objective of this work is to make the numerical analysis of the optimal control problem which is governed by the elliptic variational inequality (1.3) by proving the convergence of a discrete solution to the continuous optimal control problems. In Section 2, we establish the discrete elliptic variational inequality (2.3) which is the discrete formulation of the continuous elliptic variational inequality (1.3), and we obtain that these discrete problems have unique solutions for all positive . Moreover, on the adequate functional spaces these solutions are convergent when to the solutions of the continuous elliptic variational inequality (1.3). In Section 3, we define the discrete optimal control problem (3.2) corresponding to continuous optimal control problem (1.5). We prove the existence of a discrete solution for the optimal control problem () for each parameter and we obtain the convergence of this family with its corresponding discrete state system to the continuous optimal control with the corresponding continuous state system of the problem (). 2 Discretization of the problem (S) Let a bounded polygonal domain; a positive constant and a regular triangulation with Lagrange triangles of type 1, constituted by affine-equivalent finite elements of class over being the parameter of the finite element approximation which goes to zero [12, 18]. We take equal to the longest side of the triangles and we can approximate the sets and by: where is the set of the polynomials of degree less than or equal to in the triangle . Let be the corresponding linear interpolation operator and a constant (independent of the parameter ) such that : The discrete variational inequality formulation of the system is defined as: Find such that Let , and be, then there exist unique solution of the problem given by the elliptic variational inequality (2.3). Let , and , be the solutions of for and respectively, then we have that: there exist a constant independent of such that: if in weak, then in strong for each fixed . a) If we consider in the discrete elliptic variational inequality (2.3) we have: where is the trace operator and therefore (2.4) holds. b) As and are respectively the solutions of discrete elliptic variational inequalities (2.3) for y , we have: for . By coerciveness of we deduce: thus (2.5) holds. c) Let be. From item a) we have that , then there exist such that in weak (in strong). If we consider the discrete elliptic inequality (2.3) we have: and using that is a lower weak semi-continuous application then, when goes to infinity, we obtain that: and from uniqueness of the solution of problem , we deduce that . Now, it is easily to see that: and from the coerciveness of we obtain As in and in , by pass to the limit when in the previous inequality, we obtain Henceforth we will consider the following definitions : Given and , we have the convex combinations of two data the convex combination of two discrete solutions and we define as the associated state system which is the solution of the discrete elliptic variational inequality (2.3) for the control . Then, we have the following properties: Given the controls , we have that: a) From the definition (2.8) we get then we conclude (2.9). b) It follows from a similar method to the part a). ∎ From Lemma 2.1 we have that there exist a constant independent of such that then we conclude that there exists so that in weak as and . On the other hand, given there exist such that for each and in strong when goes to zero. Now, by considering in the discrete elliptic variational inequality (2.3) we get: and when we pass to the limit as in (2.11) by using that the bilinear form is lower weak semicontinuous in we obtain: that it is to say: and, from the uniqueness of the solution of the discrete elliptic variational inequality (1.3), we obtain that then by pass to the limit when it results that ∎ 3 Discretization of the optimal control problem Now, we consider the continuous optimal control problem which was established in (1.5). The associated discrete cost functional is defined by the following expression: and we establish the discrete optimal control problem as: Find such that where is the associated state system solution of the problem which was described for the discrete elliptic variational inequality (2.3) for a given control . Given the control , we have: b) for some constant independent of . c) The functional es a lower weakly semi-continuous application in . d) There exists a solution of the discrete optimal control problem (3.2) for all . a) From the definition of we obtain a) and b). c) Let in weak, then by using the equality we obtain that . Therefore, we have d) It follows from . ∎ If the continuous state system has the regularity then we have the following estimations : where ’s are constants independents of . and then (3.3) holds. b) From the definitions of and , it results: Following the idea given in we define an open problem: Given the controls , Remark 1: We have that . Remark 2: The equivalent inequality for the continuous optimal control problem is true, that is : for all , where is the unique solution of the elliptic variational inequality when we consider instead of , and is the unique solution of the elliptic variational inequality when we consider instead of . and therefore, the uniqueness for the discrete optimal control problem in the theorem holds. Let be the continuous state system associated to the optimal control which is the solution of the continuous distributed optimal control problem (1.5). If, for each , we choose an optimal control which is the solution of the discrete distributed optimal control problem (3.2) and its corresponding discrete state system , we obtain that: Then, if we consider and his corresponding associated state system, it results that: From the Lemma 2.1 we have that , then we can obtain: If we consider in the inequality (2.3) for , we obtain: and from the coerciveness of the application we have that and in consequence . Now we can say that there exist and such that in weak (in strong), and in weak when . Then, and in i.e., . Let given , there exist such that in strong when . Then, if we consider the variational elliptic inequality (2.3) for we have: Taking into account that the application is a lower weak semi-continuous application in and by pass to the limit when goes to zero in (3.10) we obtain that: and by the uniqueness of the solution of the problem given by the elliptic variational inequality (1.3), we deduce that . Finally, the norm on is a lower semi-continuous application in the weak topology, then we can prove that: and because the uniqueness of the optimal problem (1.5), it results that and Now, if we consider in the elliptic variational inequality (1.3) for the control and we define , we have that: and by consider for in the inequality (2.3) we obtain: and then by the coerciveness of we get When we pass to the limit as in (3.11) and by using the strong convergence of to on and the weak convergence of to on , we have: The strong convergence of the optimal controls to is obtained by using Theorem 3.1 and weakly on , i.e. then and therefore . 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burr distribution 3 parameters December 4, 2020 – Posted in: Uncategorized The two parameter BX has several types of distribution like Rayleigh (R) when \((\theta = 1)\) and Burr type X distribution with one parameter (BX1) when \((\lambda =1)\).BX1 has been studied by some authors, for example: Ahmad Sartawi and Abu-Salih (), Jaheen (), Jaheen (), Ahmad et al. The dBurr(), pBurr(), qBurr(),and rBurr() functions serve as wrappers of the dparetoIV, pparetoIV, qparetoIV, and rparetoIV functions in the VGAM package. RANI (1997). Generate sample data from a Burr distribution with scale parameter 0.5 and shape parameters 2 and 5. Need help with a homework or test question? Comments? The exact expression of the expected Fisher information matrix of the parameters in the distribution is obtained. Tadikamalla, Pandu R. (1980), “A Look at the Burr and Related Distributions”, International Statistical Review 48 (3): 337–344. It addresses the problem of estimating the three-parameter Burr XII distribution and its doubly truncated It can fit a wide range of empirical data, and is used in various fields such as finance, hydrology, and reliability to model a variety of data types. Need to post a correction? It addresses the problem of estimating the three-parameter Burr XII distribution and its doubly truncated Please post a comment on our Facebook page. Applied Mathematics and Information Sciences, 11, no. The shape of a Burr distribution associated with or is contingent on the values of the shape parameters (and ), which can be determined by simultaneously solving equations (16) and (17) from [2, p. 2211] for given values of skew and kurtosis. Also, an approximation based on Lindley is used to obtain the Bayes estimator. Burr Type III distribution has two categories: First a two-parameter distribution which has two shape parameters and second a three-parameter distribution which has a scale and two shape parameters. for 2-parameter and c, k and s for 3-parameter of Burr Type XII distribution with co mplete and censored data using two methods which include MLE and EM algorithm Tables 7 and 8 show the estimated Burr distribution parameters for Glen Osmond and South Road link travel time data sets. Parameter estimates We can apply a bootstrap to estimate the uncertainty in the parameters: Density, distribution function, quantile function and random generation for the Burr distribution with a and k two parameters. Based on this family, we define a new four-parameter extension of the Burr III distribution. A five-parameter distribution, the beta Burr XII, is useful for modeling lifetime data. A.CHATURVEDI,U. CLICK HERE! ", https://en.wikipedia.org/w/index.php?title=Burr_distribution&oldid=980569186, Creative Commons Attribution-ShareAlike License, This page was last edited on 27 September 2020, at 07:20. The Burr Type III distribution has been applied in the study of income, wage and wealth. Online Tables (z-table, chi-square, t-dist etc. Johnson, N.L. Burr type III lower record values are generated using the inverse cdf, X i = (u i −1/k − 1) −1/c, where u i is the uniformly distributed random variate. It is widely recognised that the three-parameter Burr XII distribution has Weibull distribution as a limiting case as α →+∞ with φ / α1/τ = θ remaining finite (which implies that φ →+∞ simultaneously); see, for example, Rodriguez (1977), Watkins (1999) and Shao (2000). Feroze, N. & Aslam, M. (2013) “Maximum Likelihood Estimation of Burr Type V Distribution under Left Censored Samples.” WSEAS Transactions on Mathematics. Other forms of this distribution have very little research associated with them. It can have decreasing, unimodal and decreasing-increasing-decreasing hazard rate function. Vol. Estimation procedures for a family of density functions representing various life-testing models. Figures 1-4 gives the pdf plot for three parameter Burr type XII distribution and Lomax distribution for different values of parameters. et. Density function, distribution function, quantile function, random generation,raw moments and limited moments for the Burr distribution withparameters shape1, shape2 and scale. y1label Correlation Coefficient x1label R burr type 3 ppcc plot y let r = shape1 let k = shape2 justification center move 50 6 text Rhat = ^r (R = ^rsav), Khat = ^k (K = ^ksav) move 50 2 text Maximum PPCC = ^maxppcc . , See Kleiber and Kotz (2003), Table 2.4, p. 51, "The Burr Distributions. S= Q(6 8 “Continuous Univariate Distributions”. This paper identifies the characteristics of three-parameter Burr Type XII distribution and discusses its utility in survivorship applications. Both functions support censored data for Burr distribution. Table 7. ). Expectation-maximization (EM) algorithm method is selected in this paper to estimate the two- and three-parameter Burr Type III distributions. Descriptive Statistics: Charts, Graphs and Plots. , The Burr Type XII distribution is a member of a system of continuous distributions introduced by Irving W. Burr (1942), which comprises 12 distributions. al (2013) say about the Type V that “Many properties of the parameters of the distribution under different estimation procedures are still to be revealed.”, References: The cdf is: The Burr distribution is very similar (and is, in some cases, the same as) many other distributions such as: In 1941, Burr introduced twelve cumulative distribution functions that could be fit to real life data. rdrr.io Find an R package R language docs Run R in your browser R Notebooks. When the fourth parameter, γ, equals zero, it gives a three parameter (c,k,α) distribution. let y = burr type 3 random numbers for i = 1 1 200 let y = 10*y let amax = maximum y . A summary of the models is provided in Table 22.3.For each distribution model, the table lists the parameters in the order in which they appear in the signature of the functions or subroutines that accept distribution parameters as input or output arguments. In my papers the probability density for a burr distribution is given as. It is also known as the Singh–Maddala distribution and is one of a number of different distributions sometimes called the "generalized log-logistic distribution". A set of predefined distribution models is provided with the SEVERITY procedure. The Burr distribution is a three-parameter family of distributions on the positive real line. The Burr (Type XII) distribution has probability density function:, When c = 1, the Burr distribution becomes the Pareto Type II (Lomax) distribution. Visually and using the previous statistics, it seems that the Burr distribution seems the preferred one among the candidates we chose to explore. It is suitable to fit lifetime data since it has flexible shape and controllable scale parameters. The Burr is the distribution of the random variable s (X/(1 - X))^(1/b), where X has a beta distribution with parameters 1 and a. The pdf for the Burr XII distribution is: parameter Burr type XII distribution under the failure-censored plan. 2, John Wiley & Sons, New York, NY, USA, 2nd edition. The Burr distribution is a special case of the Pareto(IV) distribution where the location parameter is equal 0 and inequality parameter is equal to 1/g, Brazauskas (2003). 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Should 8th graders take algebra?Space and Astronomy Stronger math students take algebra in eighth grade, and although they indeed may benefit academically from the course, that does not mean that weaker students will also benefit from taking algebra earlier. What math do most 8th graders take? The primary strands for an 8th-grade math curriculum are number sense and operations, algebra, geometry, and spatial sense, measurement, and data analysis and probability. While these math strands might surprise you, they are all critical lessons for an 8th-grade math curriculum. What grade do most kids take algebra? In many schools today, algebra in the eighth grade is the norm, and students identified by some predetermined standard can complete the course in seventh grade. Algebra courses are even stratified as “honors” algebra and “regular” algebra at both of these grade levels. What is 8th grade algebra? Grade 8 Algebra is a high school level Algebra 1 course, and is the first course on their growth in upper level mathematics. The fundamental purpose of this course is to formalize and extend the mathematics that students learned through mastery of the middle school standards. Is taking geometry in 8th grade good? The concepts taught in geometry in the 8th grade math class are foundational for future understanding of geometry concepts. If students are not able to grasp these concepts, they will struggle in future math classes. This is the first year, for example, that students will make proofs to prove that something is true. Is algebra harder than geometry? Is geometry easier than algebra? Geometry is easier than algebra. Algebra is more focused on equations while the things covered in Geometry really just have to do with finding the length of shapes and the measure of angles. Can 8th graders take algebra 1? Algebra. Students taking Algebra 1 in eighth grade likely completed a pre-algebra course — or at minimum, a general math course introducing basic algebraic ideas — in seventh grade. Can 8th graders take algebra 2? Algebra 2 in 8th grade is actually not that advanced. A student who takes Algebra 2 in 8th grade is above average, but not necessarily college material. What is 7th grade math? In 7th grade, students will fully understand how to interpret and compute all rational numbers. They can add, subtract, multiply, and divide all decimals and fractions, as well as represent percents. How do you skip a grade? Requirements to Skip a Grade - A Written Request. Put your request for skipping a grade in writing to the school principal and keep a copy. … - Expert Guidance. Make sure that legitimate requirements are being used in considering your request. … - Academic Achievement. … - Emotional Readiness. … - Student Acceptance. … - Need for Change. What do 8th graders learn in math? Students continue to develop their understanding of patterns, including those that involve integers. They use Algebraic notation, such as, s = d/t, to represent the relationship between speed, distance and time. They solve Algebraic equations involving multiple terms, integers and decimal numbers. Can 7th graders take algebra 1? Seventh graders are capable of Algebra 1 or even Geometry, depending on how well they have prepared. It’s not the age, but how well you have prepared them. If the child is going to take a College Major related to Math or Math skills required, then try to take Algebra in 7th. grade at least. Is my child ready for algebra? According to the National Council of Teachers of Mathematics (NCTM), some indicators of algebra readiness include: The ability to use properties such as commutativity, associativity, and distributivity. E.g., knowing that 5 + 9 is equal to 9 + 5 (commutativity) Which is harder algebra or pre-algebra? Prealgebra introduces algebra concepts and takes each one slower and therefore does not cover as much material as a standard Algebra I course. Some parents find it is just as easy to take a regular Algebra I course and do it in two years, especially if the student is in the 6th or 7th grade. What math is 6th grade? The major math strands for a sixth-grade curriculum are number sense and operations, algebra, geometry, and spatial sense, measurement, and functions, and probability. What grade do you learn algebra? Algebra is the culmination of most elementary & middle school math programs. Typically, algebra is taught to strong math students in 8th grade and to mainstream math students in 9th grade. What are the levels of algebra? Algebra is divided into different sub-branches such as elementary algebra, advanced algebra, abstract algebra, linear algebra, and commutative algebra. - Compaction in the Rock Cycle: Understanding the Process Behind Sedimentary Rock Formation - Crystallization in the Water Cycle: A Fundamental Process in Water Distribution and Purification - Understanding Crystallization in the Rock Cycle: A Fundamental Process in Rock Formation - SQL Server to Google Maps - Stereo-pair Image Registration - Extracting Lat/Lng from Shapefile using OGR2OGR/GDAL - Constructing query in Nominatim - In Ogr2OGR: what is SRS? - Identifying port numbers for ArcGIS Online Basemap? - Remove unwanted regions from map data QGIS - Waiting for Vector & WFS loading - Adding TravelTime as Impedance in ArcGIS Network Analyst? - Listing total number of features into an ArcGIS Online feature pop-up - Criteria for cartographic capacity

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Make sure your hypothesis is a specific statement relating to a single experiment. The way that you structure your research questions; that is, the way that you write out your research questions will vary depending on the type of research question you are trying to answer [see the article: Administrators who provide wellness programs for their employees receive higher employee ratings on selected leadership qualities than administrators who do not provide wellness programs. To learn more about these three types of quantitative research question i. Sometimes, the dependent variable needs to be broken into two parts around the group s you are interested in so that the research question flows. The examples illustrate the difference between the use of a single group e. Identified the variables in the project. Does the amount of calcium in the diet of elementary school children effect the number of cavities they have per year? We make an "educated guess. Describe the purpose of each and understand the importance of a well-developed question or hypothesis. Independent and dependent variables A variable is not only something that we measure. We could describe factors relating to the make-up of these Facebook users, quantifying how many or what proportion of these university students were male or female, or what their average age was. Dependent variable 1st; group 2nd: In the second example, the research question is not only interested in what the factors influencing career choices are, but which of these factors are the most important. The first two examples highlight that while the name of the dependent variable is the same, namely daily calorific intake, the way that this dependent variable is written out differs in each case. Does the question flow? Sometimes the number of sources you find will help you discover whether your research question is too broad, too narrow, or okay? How to structure quantitative research questions ]. These three approaches to examining the variables you are interested in i. A complete hypothesis should include: As a general rule, we would suggest using hypotheses rather than research questions in these cases. It builds upon previously accumulated knowledge e. The following video may be helpful in learning how to choose appropriate keywords and search online databases: This question may allow the researcher to collect data but does not lend itself to collecting data that can be used to create a valid argument because the data is just factual information. How many calories are consumed per day by American men and women? If you feel like the research questions are no more than a repetition of the research hypotheses, it is often better to include only one or the other i. Who will it help and how? Research Questions, discusses how to choose whether to use a hypothesis or a question when creating a research project. Name of the dependent variable How the dependent variable is written out Daily calorific intake How many calories do American men and women consume per day? We use the word groups of variables because both categorical and continuous variables include additional types of variables. When performing quantitative analysis on the data you collect during the dissertation process, you need to understand what type of categorical or continuous variables you are measuring. Is it a hot topic, or is it becoming obsolete? What are the effects of intervention programs in the elementary schools on the rate of childhood obesity among 3rd - 6th grade students? If so, you will be using a hypothesis.Often, one of the trickiest parts of designing and writing up any research paper is writing the hypothesis. How to structure quantitative research questions. Writing out the problem or issues you are trying to address in the form of a complete research question. In this article, Select the appropriate structure for the chosen type of quantitative research question, based on the variables and/or groups involved. This section of the article briefly discusses the difference between these three types of quantitative research question. What variables are you trying to measure, manipulate and/or control? Dissertations that are based on a quantitative research design attempt to answer at least one quantitative research question. These hypothesis. A complete hypothesis should include: the variables, the population, and the predicted relationship between the variables. Planning My Research Question or Hypothesis – This resources contains a link to a PowerPoint presentation and a series of tutorials that contain examples and tips for writing research questions and hypotheses. Research Question Why Intro Develop a Theory Your Answer Intro Identify Variables (if applicable) more variables.” “A hypothesis can be defined as a tentative explanation of the research To be either writing-questions, or. Developing Research Questions: Hypotheses and Variables Common Sources of Research Questions Professors Textbooks Databases This is the beginning of a good research question. Topics for research As of this writing, two useful search engines are Ingenta, Galaxy and Google.Download

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It's possible to use any letter you opt for. The answer, in reality, lies somewhere in the middle. The context affects a number of the above definitions and terminology. A term is a mathematical expression which might form a separable portion of an equation, a series, or a different expression. research paper In many instances, introducing heuristics may increase the random algorithm. Details of What Is a Term in Math In some cases, you are going to be given one factor of a massive expression and you're going to be need to discover the remainder of the ones. So, now you may observe how a notion is translated in specific contexts. Therefore, there are many terms in the expression. The Secret to What Is a Term in Math And, due to the problem we just did before, needless to say, we are aware that number will be two. It isn't always a great idea to use patterns because in cases like exponents they might not be complete patterns for every single scenario, but the proof still holds. Sometimes getting the correct data might be the hardest aspect of a project, especially if you are trying to do something new. These substances are called indicators. You might have read that you can place a formula in a column, somewhat like a spreadsheet. Create it like a user-defined function and you might then utilize it in your formulas. Again, due to the Order of Operations that is presented in a subsequent lesson, the exponent has to be simplified before you do anything with the negative sign. It is helpful to have a chart for those roots which is precisely why we include the charts in the worksheets above. It is simply the selection of input variables. Excel has another function called LOG that's a different type of logarithm. When it has to do with relevance, the sequence can be both an incrementing price and a few of the value can be set aside to declare extra qualities in what's being defined. In addition, a variable may be constant in the event the issue specifically tells you what the variable equals. While structural equality can be checked with no understanding of the significance of the symbols, semantic equality may not. So, now you may observe how a notion is translated in specific contexts. Therefore, there are many terms in the expression. Flipping this lesson could work nicely. Let's look at a great example. Let's look at an outstanding example. It's possible to still do the mathematical operations to locate their average. The top and bottom bases just need to be the exact same. The conventional base is 10, and that's the base that's set in calculators.

ch 3000 mass transfer Popular in Department This 36 page Class Notes was uploaded by Adam on Saturday June 4, 2016. The Class Notes belongs to ch 3000 at Clemson University taught by in Summer 2016. Since its upload, it has received 36 views. Reviews for ch 3000 mass transfer Report this Material What is Karma? Karma is the currency of StudySoup. You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more! Date Created: 06/04/16 3. Molecular mass transport 3.1 Introduction to mass transfer 3.2 Properties of mixtures 3.2.1 Concentration of species 3.2.2 Mass Averaged velocity 3.3 Diffusion flux 3.3.1 Pick’s Law 3.3.2 Relation among molar fluxes 3.4 Diffusivity 3.4.1 Diffusivity in gases 3.4.2 Diffusivity in liquids 3.4.3 Diffusivity in solids 3.5 Steady state diffusion 3.5.1 Diffusion through a stagnant gas film 3.5.2 Pseudo – steady – state diffusion through a stagnant gas film. 3.5.3 Equimolar counter diffusion. 3.5.4 Diffusion into an infinite stagnant medium. 3.5.5 Diffusion in liquids 3.5.6 Mass diffusion with homogeneous chemical reaction. 3.5.7 Diffusion in solids 3.6 Transient Diffusion. 3.1 Introduction of Mass Transfer When a system contains two or more components whose concentrations vary from point to point, there is a natural tendency for mass to be transferred, minimizing the concentration differences within a system. The transport of one constituent from a region of higher concentration to that of a lower concentration is called mass transfer. The transfer of mass within a fluid mixture or across a phase boundary is a process that plays a major role in many industrial processes. Examples of such processes are: (i) Dispersion of gases from stacks (ii) Removal of pollutants from plant discharge streams by absorption (iii) Stripping of gases from waste water (iv) Neutron diffusion within nuclear reactors 1 (v) Air conditioning Many of air daybyday experiences also involve mass transfer, for example: (i) A lump of sugar added to a cup of coffee eventually dissolves and then eventually diffuses to make the concentration uniform. (ii) Water evaporates from ponds to increase the humidity of passingair stream (iii) Perfumes presents a pleasant fragrance which is imparted throughout the surrounding atmosphere. The mechanism of mass transfer involves both molecular diffusion and convection. 3.2 Properties of Mixtures Mass transfer always involves mixtures. Consequently, we must account for the variation of physical properties which normally exist in a given system. When a system contains three or more components, as many industrial fluid streams do, the problem becomes unwidely very quickly. The conventional engineering approach to problems of multicomponent system is to attempt to reduce them to representative binary (i.e., two component) systems. In order to understand the future discussions, let us first consider definitions and relations which are often used to explain the role of components within a mixture. 3.2.1 Concentration of Species: Concentration of species in multicomponent mixture can be expressed in many ways. For species A, mass concentration denoted by isAdefined as the mass of A,m A per unit volume of the mixture. m A (1) A V The total mass concentration density is the sum of the total mass of the mixture in unit volume: i i where iis the concentration of species i in the mixture. 2 Molar concentration of, A,AC is defined as the number of moles of A present per unit volume of the mixture. By definition, mass of A es molecular weight of A (2) A M A Therefore from (1) & (2) C n A A A V M A For ideal gas mixtures, pAV A R T [ from Ideal gas law PV = nRT] n A pA A V RT where pA is the partial pressure of species A in the mixture. V is the volume of gas, T is the absolute temperature, and R is the universal gas constant. The total molar concentration or molar density of the mixture is given by C C i 3.2.2 Velocities In a multicomponent system the various species will normally move at different velocities; and evaluation of velocity of mixture requires the averaging of the velocities of each species present. If I is the velocity of species i with respect to stationary fixed coordinates, then massaverage velocity for a multicomponent mixture defined in terms of mass concentration is, 3 i i i i i i i i By similar way, molaraverage velocity of the mixture * is C i i * C For most engineering problems, there will be title difference in * and and so the mass average velocity, , will be used in all further discussions. The velocity of a particular species relative to the massaverage or molar average velocity is termed as diffusion velocity (i.e.) Diffusion velocity i The mole fraction for liquid and solid mixture,Ax ,and for gaseous mixtures,A y , are the molar concentration of species A divided by the molar density of the mixtures. C A x A (liquids and solids) C C y A A (gases). C The sum of the mole fractions, by definition must equal 1; x i 1 (i.e.) i i by similar way, mass fraction of A in mixture is; A 4 5 10. The molar composition of a gas mixture at 273 K and 1.5 * 10 Pa is: O 7% 2 CO 10% CO 215% N 2 68% Determine a) the composition in weight percent b) average molecular weight of the gas mixture c) density of gas mixture d) partial pressure of 2. Calculations: Let the gas mixture constitutes 1 mole. Then O 2 = 0.07 mol CO = 0.10 mol CO 2= 0.15 mol N 2 = 0.68 mol Molecular weight of the constituents are: O 2 = 2 * 16 = 32 g/mol CO = 12 + 16 = 28 g/mol CO 2= 12 + 2 * 16 = 44 g/mol N 2 = 2 * 14 = 28 g/mol Weight of the constituents are: (1 mol of gas mixture) O 2 = 0.07 * 32 = 2.24 g CO = 0.10 * 28 = 2.80 g CO 2= 0.15 * 44 = 6.60 g N 2 = 0.68 * 28 = 19.04 g Total weight of gas mixture = 2.24 + 2.80 + 6.60 + 19.04 = 30.68 g Composition in weight percent: 2.24 O 2 *100 7.30% 30.68 5 2.80 CO 30.68*100 9.13% CO 6.60*100 21.51% 2 30.68 19.04 N 2 *100 62.06% 30.68 Weight of gas mixture Average molecular weight of the gas mixture Number of moles M 30.68 30.68 g mol 1 Assuming that the gas obeys ideal gas law, PV = nRT n P V RT n molardensity m V Therefore, density (or mass density) m M Where M is the molecular weight of the gas. PM 1.5 * 105 * 30.68 3 Density m M kg m RT 8314 * 273 = 2.03 kg/m Partial pressure of 2 = [mole fractio 2of O ] * total pressure 7 5 * 1.5 * 10 100 5 = 0.07 * 1.5 * 10 = 0.105 * 10 Pa 3.3 Diffusion flux Just as momentum and energy (heat) transfer have two mechanisms for transportmolecular and convective, so does mass transfer. However, there are convective fluxes in mass transfer, even on a molecular level. The reason for this is that in mass transfer, whenever there is a driving force, there is always a net 6 movement of the mass of a particular species which results in a bulk motion of molecules. Of course, there can also be convective mass transport due to macroscopic fluid motion. In this chapter the focus is on molecular mass transfer. The mass (or molar) flux of a given species is a vector quantity denoting the amount of the particular species, in either mass or molar units, that passes per given increment of time through a unit area normal to the vector. The flux of species defined with reference to fixed spatial coordinates,AN is A A A (1) This could be written interms of diffusion velocity of A, (i.A., ) and average velocity of mixture, , as N C ( ) C A A A A (2) By definition C i i * i C Therefore, equation (2) becomes C A A A (A ) C i i C i y A C i i i For systems containing two components A and B, A A (A ) y A(C A A C B B) A A ) y A (N A N B ) N C ( ) y N A A A A (3) The first term on the right hand side of this equation is diffusional molar flux of A, and the second term is flux due to bulk motion. 3.3.1 Fick’s law: An empirical relation for the diffusional molar flux, first postulated by Fick and, accordingly, often referred to as Fick’s first law, defines the diffusion of 7 component A in an isothermal, isobaric system. For diffusion in only the Z direction, the Fick’s rate equation is d C A A AB d Z where D ABis diffusivity or diffusion coefficient for component A diffusing through component B, and dC /AdZ is the concentration gradient in the Zdirection. A more general flux relation which is not restricted to isothermal, isobasic system could be written as d y A A D AB (4) d Z using this expression, Equation (3) could be written as d y A AB A y AN (5) d Z 3.3.2 Relation among molar fluxes: For a binary system containing A and B, from Equation (5), A A y AN or A A y A N (6) Similarly, B B y B N (7) Addition of Equation (6) & (7) gives, J J N N (y y )N A B A B A B (8) By definition N = NA + NB and A + B = 1. Therefore equation (8) becomes, A B J + J = 0 A B J = J 8 d y d y AB A CD BA B (9) d z d Z From y A+ y B= 1 dy = dy A B Therefore Equation (9) becomes, AB D BA (10) This leads to the conclusion that diffusivity of A in B is equal to diffusivity of B in A. 3.4 Diffusivity Fick’s law proportionality, D , AB known as mass diffusivity (simply as diffusivity) 2 or as the diffusion coefficient. D hAB the dimension of L / t, identical to the fundamental dimensions of the other transport properties: Kinematic viscosity, = ( / ) in momentum transfer, and thermal diffusivity, (= k / C ) in he transfer. Diffusivity is normally reported in cm / sec; the SI unit being m / sec. Diffusivity depends on pressure, temperature, and composition of the system. In table, some values of D areABiven for a few gas, liquid, and solid systems. Diffusivities of gases at low density are almost composition independent, incease with the temperature and vary inversely with pressure. Liquid and solid diffusivities are strongly concentration dependent and increase with temperature. General range of values of diffusivity: –6 5 2 Gases : 5 X 10 1 X 10 m / sec. Liquids : 10 10 m / sec. Solids : 5 X 10 1 X 10 m / sec. In the absence of experimental data, semitheoretical expressions have been developed which give approximation, sometimes as valid as experimental values, due to the difficulties encountered in experimental measurements. 9 3.4.1 Diffusivity in Gases: Pressure dependence of diffusivity is given by 1 AB (for moderate ranges of pressures, upto 25 atm). p And temperature dependency is according to 3 D T 2 AB Diffusivity of a component in a mixture of components can be calculated using the diffusivities for the various binary pairs involved in the mixture. The relation given by Wilke is 1 D 1mixture y2 y3 ........... n D D D 12 13 1n Where D is the diffusivity for component 1 in the gas mixture; D is the 1mixture 1n diffusivity for the binary pair, component 1 diffusing through component n; and y is the mole fraction of component n in the gas mixture evaluated on a n component –1 – free basis, that is y2 2 y2 y 3.......y n 9. Determine the diffusivity of Co 2(1), O 22) and N 2) in a gas mixture having the composition: Co 2: 28.5 %, O 2: 15%, N 256.5%, The gas mixture is at 273 k and 1.2 * 10 Pa. The binary diffusivity values are given as: (at 273 K) D P = 1.874 m Pa/sec 12 2 D 13= 1.945 m Pa/sec D 23= 1.834 m Pa/sec Calculations: Diffusivity of C 2 in mixture 10 1 D 1m y 2 y 3 D 12 D 13 y y2 0.15 0.21 where 2 y y 0.15 0.565 2 3 y3 0.565 y3 0.79 y 2 y 3 0.150.565 1 D 1m P Therefore 0.21 0.79 1.874 1.945 = 1.93 m .Pa/sec Since P = 1.2 * 10 Pa, 1.93 5 2 D1m 5 1.61*10 m sec 1.2*10 Diffusivity of 2 in the mixture, 1 D2m y1 y3 D D 21 23 y 1 0.285 Where y1 0.335 y1 y 3 0.285 0.565 (mole fraction on2 free bans). and y y3 3 0.565 0.665 y 1 y 3 0.285 0.565 and D 21 = D 12= 1.874 m .Pa/sec Therefore 1 D 2mP 0.335 0.665 1.874 1.834 2 = 1.847 m .Pa/sec 11 D m 1.847 1.539*10 5 m sec 2 1.2*10 5 By Similar calculations Diffusivity of N i 2the mixture can be calculated, and is found to be, D 3m.588 * 10 m /sec. 2 3.4.2 Diffusivity in liquids: Diffusivity in liquid are exemplified by the values given in table … Most of these values are nearer to 10 cm / sec, and about ten thousand times shower than those in dilute gases. This characteristic of liquid diffusion often limits the overall rate of processes accruing in liquids (such as reaction between two components in liquids). In chemistry, diffusivity limits the rate of acidbase reactions; in the chemical industry, diffusion is responsible for the rates of liquidliquid extraction. Diffusion in liquids is important because it is slow. Certain molecules diffuse as molecules, while others which are designated as electrolytes ionize in solutions and diffuse as ions. For example, sodium chloride (NaCl), diffuses in water as ions Na and Cl. Though each ions has a different mobility, the electrical neutrality of the solution indicates the ions must diffuse at the same rate; accordingly it is possible to speak of a diffusion coefficient for molecular electrolytes such as NaCl. However, if several ions are present, the diffusion rates of the individual cations and anions must be considered, and molecular diffusion coefficients have no meaning. Diffusivity varies inversely with viscosity when the ratio of solute to solvent ratio exceeds five. In extremely high viscosity materials, diffusion becomes independent of viscosity. 3.4.3 Diffusivity in solids: Typical values for diffusivity in solids are shown in table. One outstanding characteristic of these values is their small size, usually thousands of time less than those in a liquid, which are inturn 10,000 times less than those in a gas. Diffusion plays a major role in catalysis and is important to the chemical engineer. For metallurgists, diffusion of atoms within the solids is of more importance. 3.5 Steady State Diffusion 12 In this section, steadystate molecular mass transfer through simple systems in which the concentration and molar flux are functions of a single space coordinate will be considered. In a binary system, containing A and B, this molar flux in the direction of z, as given by Eqn (5) is [section 3.3.1] d y A A CD AB d z y A (N A N )B (1) 3.5.1 Diffusion through a stagnant gas film The diffusivity or diffusion coefficient for a gas can be measured, experimentally using Arnold diffusion cell. This cell is illustrated schematically in figure. figure The narrow tube of uniform cross section which is partially filled with pure liquid A, is maintained at a constant temperature and pressure. Gas B which flows across the open end of the tub, has a negligible solubility in liquid A, and is also chemically inert to A. (i.e. no reaction between A & B). Component A vaporizes and diffuses into the gas phase; the rate of vaporization may be physically measured and may also be mathematically expressed interms of the molar flux. Consider the control volume S z, where S is the cross sectional area of the tube. Mass balance on A over this control volume for a steadystate operation yields [Moles of A leaving at z + z] – [Moles of A entering at z] = 0. (i.e.) A z z S N A z 0. (1) Dividing through by the volume, SZ, and evaluating in the limit as Z approaches zero, we obtain the differential equation 0 (2) d z This relation stipulates a constant molar flux of A throughout the gas phase from Z1 to Z2. A similar differential equation could also be written for component B as, 13 d N d Z and accordingly, the molar flux of B is also constant over the entire diffusion path from z 1and z 2. Considering only at plane z ,1and since the gas B is insoluble is liquid A, we realize that N , the net flux of B, is zero throughout the diffusion path; accordingly B B is a stagnant gas. From equation (1) (of section 3.5) d y A y (N N ) A AB d z A A B Since N B= 0, d y A A AB d z y AN A Rearranging, CD d y A AB A (3) 1y A d z This equation may be integrated between the two boundary conditions: at z = z Y = Y 1 A A1 And at2z = z A A2 Y = y Assuming the diffusivity is to be independent of concentration, and realizing that N Ais constant along the diffusion path, by integrating equation (3) we obtain Z2 yA2 d y A CD AB A Z y 1y A 1 A1 CD AB 1 y A2 A ln (4) Z 2 Z 1 1y A1 The log mean average concentration of component B is defined as 14 y y y B2 B1 y B2 ln y B1 Since y B 1 y A , (1y A2 ) (1y A1) y A1 y A2 y B,lm y A2 y A2 (5) ln y A1 ln y A1 Substituting from Eqn (5) in Eqn (4), A CD AB (y A1y A2 ) (6) Z 2 z 1 y B,lm For an ideal gas C n p , and V R T for mixture of ideal gasesy p A A P Therefore, for an ideal gas mixture equation. (6) becomes N D AB (p A1 p A2) A RT(z z ) p 2 1 B,lm This is the equation of molar flux for steady state diffusion of one gas through a second stagnant gas. Many masstransfer operations involve the diffusion of one gas component through another nondiffusing component; absorption and humidification are typical operations defined by these equation. The concentration profile (y Avs. z) for this type of diffusion is shown in figure: Figure 15 12. Oxygen is diffusing in a mixture of oxygennitrogen at 1 std atm, 25C. Concentration of oxygen at planes 2 mm apart are 10 and 20 volume % respectively. Nitrogen is nondiffusing. (a) Derive the appropriate expression to calculate the flux oxygen. Define units of each term clearly. (b) Calculate the flux of oxygen. Diffusivity of oxygen in nitrogen = 1.89 * 10 2 m /sec. Solution: Let us denote oxygen as A and nitrogen as B. Flux of A (i.e.) N Ais made up of two components, namely that resulting from the bulk motion of A (i.e.), Nx Aand that resulting from molecular diffusion A : N A NxA J A (1) From Fick’s law of diffusion, d C A JA D AB (2) d z Substituting this equation (1) d C N A Nx A D AB A (3) d z Since N = N A N Bnd x A C AC equation (3) becomes C A dC A N A N A NB C DAB d z Rearranging the terms and integrating between the planes between 1 and 2, d z C dC A C A2 (4) cD AB A1 N A C A N A N B Since B is non diffusing B = 0. Also, the total concentration C remains constant. Therefore, equation (4) becomes 16 z C A2 dC A CD AB CA1 N CA N C A A 1 C C A2 N ln C C A A1 Therefore, CD AB C C A2 N A ln (5) z C C A1 Replacing concentration in terms of pressures using Ideal gas law, equation (5) becomes D AB P t P t P A2 N A RTz lnP P (6) t A1 where D ABmolecular diffusivity of A in B P T total pressure of system R = universal gas constant T = temperature of system in absolute scale z = distance between two planes across the direction of diffusion P A1partial pressure of A at plane 1, and P A2partial pressure of A at plane 2 Given: D AB1.89 * 10 m /sec P = 1 atm = 1.01325 * 10 N/m 2 t T = 25C = 273 + 25 = 298 K z = 2 mm = 0.002 m P = 0.2 * 1 = 0.2 atm (From Ideal gas law and additive pressure rule) A1 P A20.1 * 1 = 0.1 atm Substituting these in equation (6) 5 5 N 1.89*10 1.01325*10 ln 1 0.1 A 8314 298 0.002 1 0.2 –5 2 = 4.55 * 10 kmol/m .sec 17 3.5.2 Psuedo steady state diffusion through a stagnant film: In many mass transfer operations, one of the boundaries may move with time. If the length of the diffusion path changes a small amount over a long period of time, a pseudo steady state diffusion model may be used. When this condition exists, the equation of steady state diffusion through stagnant gas’ can be used to find the flux. figure If the difference in the level of liquid A over the time interval considered is only a small fraction of the total diffusion path, an0 t – t is relatively long period of time, at any given instant in that period, the molar flux in the gas phase may be evaluated by C D AB (y A1 y A2 ) A zy (1) B,lm where z equals z 2– z 1 the length of the diffusion path at time t. The molar flux N A is related to the amount of A leaving the liquid by A,L d z A (2) M A d t A,L where is the molar density of A in the liquid phase M A under Psuedo steady state conditions, equations (1) & (2) can be equated to give d z C D (y y ) A,L AB A1 A2 (3) M A d t z y B,lm Equation. (3) may be integrated from t = 0 to t and from z = z t0to z = t as: t A,Ly B,lm M A Z t zdz t0 C D AB (y A1 y A2) Zt0 yielding 18 A,L yB,lm M A z t z t0 (4) CD AB (y A1 y A2) 2 This shall be rearranged to evaluate diffusivity D as, AB 2 2 A,L y B,lm z t z t0 AB M C (y y )t 2 A A1 A2 1. A vertical glass tube 3 mm in diameter is filled with liquid toluene to a depth of 20mm from the top openend. After 275 hrs at 39.4 C and a total pressure of 760 mm Hg the level has dropped to 80 mm from the top. Calculate the value of diffusivity. Data: vapor pressure of toluene at 39.4C = 7.64 kN / m , 2 density of liquid toluene = 850 kg/m 3 Molecular weight of toluene = 92 (C 6 6H ) 3 A,L y Blm Z Z 2 D AB t t0 figure M CAy A1 y A2 t 2 y B2 y B1 y B,l m where yB2 ln y B1 y B2 1 – y A2 y B1 1 – y A1 p A1 7.64 2 y A1 (760 mm Hg = 101.3 kN/m ) P 101.3 = 0.0754 y B1 1 – 0.0754 = 0.9246 y A2 0 y B= 1 – y A2 y 1 0.9246 0.9618 Therefore B,lm 1 ln 0.9246 5 P 1.01325*10 C RT 8314* 273 39.4 3 = 0.039 k mol /m Therefore 19 850 * 0.9618 0.08 2 0.022 D AB 92 * 0.039 * 0.0754 0 * 275 * 3600 2 –3 2 2 = 1.5262 * 10 (0.08 – 0.02 ) = 9.1572 * 10 m /sec. 3.5.3 Equimolar counter diffusion: A physical situation which is encountered in the distillation of two constituents whose molar latent heats of vaporization are essentially equal, stipulates that the flux of one gaseous component is equal to but acting in the opposite direction from the other gaseous component; that is, N = A N . B The molar flux N ,Afor a binary system at constant temperature and pressure is described by d y A A CD AB y A (N A N )B d z dC A or A D AB y A (N A N )B (1) d z with the substitution of NB = NA, Equation (1) becomes, dC A A D AB (2) d z For steady state diffusion Equation. (2) may be integrated, using the boundary conditions: at1 z = zA A1 C = C 2 AC =A2C Giving, Z C 2 A2 A z D AB dC A Z 1 C A1 from which D AB A (C A1 C A2) (3) z2z 1 20 For ideal gases, C n A pA . Therefore Equation. (3) becomes A V RT D AB A (P A1 P A2 ) (4) RT (z 2z ) 1 This is the equation of molar flux for steadystate equimolar counter diffusion. Concentration profile in these equimolar counter diffusion may be obtained from, d d z A ) 0 (Since NA is constant over the diffusion path). And from equation. (2) N D dC A A AB d z . Therefore dC A 0 . d z AB d z d 2 C or 0. d z 2 This equation may be solved using the boundary conditions to give C A C A1 zz 1 C C z z (5) A1 A2 1 2 Equation, (5) indicates a linear concentration profile for equimolar counter diffusion. 3. Methane diffuses at steady state through a tube containing helium. At point 1 the partial pressure of methane is p A155 kPa and at point 2, 0.03 m apart P = A2 15 KPa. The total pressure is 101.32 kPa, and the temperature is 298 K. At this –5 2 pressure and temperature, the value of diffusivity is 6.75 * 10 m /sec. i) calculate the flux of CH at steady state for equimolar counter 4 diffusion. ii) Calculate the partial pressure at a point 0.02 m apart from point 1. 21 Calculation: For steady state equimolar counter diffusion, molar flux is given by D AB N A p A1 pA 2 (1) RT z Therefore; 5 N 6.75*10 55 15 kmol A 8.314 * 298 * 0.03 m .sec 3.633 * 105 kmol m sec And from (1), partial pressure at 0.02 m from point 1 is: 5 3.633 * 105 6.75 * 10 55 p 8.314 * 298 * 0.02 A p A= 28.33 kPa 11. In a gas mixture of hydrogen and oxygen, steady state equimolar counter diffusion is occurring at a total pressure of 100 kPa and temperature of 20C. If the partial pressures of oxygen at two planes 0.01 m apart, and perpendicular to the direction of diffusion are 15 kPa and 5 kPa, respectively and the mass –5 2 diffusion flux of oxygen in the mixture is 1.6 * 10 kmol/m .sec, calculate the molecular diffusivity for the system. Solution: For equimolar counter current diffusion: D AB N A p A1p A2 (1) RTz where –5 2 N A= molar flux of A (1.6 * 10 kmol/m .sec): D AB= molecular diffusivity of A in B R = Universal gas constant (8.314 kJ/kmol.k) T = Temperature in absolute scale (273 + 20 = 293 K) z = distance between two measurement planes 1 and 2 (0.01 m) 22 P A1 partial pressure of A at plane 1 (15 kPa); and P A2 partial pressure of A at plane 2 (5 kPa) Substituting these in equation (1) D 1.6 * 105 AB 155 8.314 293 0.01 –5 2 Therefore, D AB 3.898 * 10 m /sec 2. A tube 1 cm in inside diameter that is 20 cm long is filled with Co2 and H2 at a total pressure of 2 atm at 0C. The diffusion coefficient of the Co2 – H 2system 2 under these conditions is 0.275 cm /sec. If the partial pressure of Co2 is 1.5 atm at one end of the tube and 0.5 atm at the other end, find the rate of diffusion for: i) steady state equimolar counter diffusion (N A= N B ii) steady state counter diffusion where N B 0.75 N Aand iii) steady state diffusion of C2 through stagnant H2 (NB = 0) d y A i) A CD AB d z y AN A NB Given N B N A d y A d C A Therefore N A C D AB D AB d z d z p A (For ideal gas mixture C A where p Ais the partial pressure of A; such that RT p A+ p B= P) d p AT Therefore N A D AB d z For isothermal system, T is constant D d p Therefore N A AB A RT d z Z 2 D AB P A2 (i.e.) N A d z d pA Z RT P 1 A1 23 D AB N A p A1 pA2 (1) RT z where Z = Z 2Z 1 2 –4 2 Given: D AB0.275 cm /sec = 0.275 * 10 m /sec ; T = 0C = 273 k 4 N 0.275 *10 1.5 * 1.01325 * 105 0.5 * 1.01325 * 105 A 8314 * 273 * 0.2 6.138 * 10 6 k mol m 2 sec Rate of diffusion = N S A Where S is surface area Therefore rate of diffusion = 6.138 * 10 * r –6 –2 2 = 6.138 * 10 * (0.5 * 10 ) = 4.821 * 10 –1 k mol/sec –3 = 1.735 * 10 mol/hr. ii) C D d y A y N N A AB d z A A B given: N B 0.75 N A d y A Therefore N A C D AB d z y A N A 0.75N A d y C D A B A 0.25 y A N A d z N 0.25 y N C D d yA A A A AB d z d y A N A d z C D AB 1 0.25 y A for constant N And C Z 2 y A2 d y N d z CD A A AB 1 0.25 y A Z 1 y A1 d x 1 ln a b x a b x b 24 N z CD 1 ln 1 0.25y A2 A AB 0.25 A y A1 4CD AB 1 0.25 y A 2 N A ln (2) z 1 0.25 y A1 Given: p 2 * 1.01325 * 10 5 C 0.0893 K mol m 3 RT 8314 * 273 p y A1 1.5 0.75 A1 P 2 p y A2 0.5 0.25 A2 P 2 Substituting these in equation (2), 4 * 0.0893 * 0.275 * 10 4 1 0.25 * 0.25 N A ln 0.2 1 0.25 * 0.75 6 kmol 7.028 * 10 2 m sec Rate of diffusion = N S = 7.028 * 10 * * (0.5 * 10 ) –2 2 A –10 = 5.52 * 10 kmol/sec = 1.987 * 10 mol/hr. d y A iii) A CD AB y A N A N B d z Given: N B 0 d y A Therefore N A CD AB y A N A d z Z 2 yA2 d y N d z CD A A Z AB y 1 y A 1 A1 CD AB 1 y A2 ln Z 1 y A1 0.0893* 0.275*10 4 1 0.25 ln 0.2 1 0.75 5 kmol 1.349 * 10 2 m .sec 25 Rate of diffusion = 1.349 8 10 * * (0.5 * 10 ) 2 = 1.059 Kmol / sec = 3.814 mol/hr 3.5.4 Diffusion into an infinite standard medium : Here we will discuss problems involving diffusion from a spherical particle into an infinite body of stagnant gas. The purpose in doing this is to demonstrate how to set up differential equations that describe the diffusion in these processes. The solutions, obtained are only of academic interest because a large body of gas in which there are no convection currents is unlikely to be found in practice. However, the solutions developed here for these problems actually represent a special case of the more common situation involving both molecular diffusion and convective mass transfer. a) Evaporation of a spherical Droplet: As an example of such problems, we shall consider the evaporation of spherical droplet such as a raindrop or sublimation of naphthalene ball. The vapor formed at the surface of the droplet is assumed to diffuse by molecular motions into the large body of stagnant gas that surrounds the droplet. Consider a raindrop, as shown in figure. At any moment, when the radius Figure Evaporation of a raindrop of the drop is r , the flux of water vapor at any distance r from the center is given 0 by d y A N A C D AB y AN A N B d r Here N = 0 (since air is assumed to be stagnant) B Therefore, d y A N A C D AB y A N A d r Rearranging, C D AB d y A N A __________ (1) 1 y A d r The flux N Ais not constant, because of the spherical geometry; decreases as the distance from the center of sphere increases. But the molar flow rate at r and r + r are the same. This could be written as, A N A r A N A r r __________ (2) where A = surface area of sphere at r or r + r. 26 2 Substituting for A = 4 r in equation (2), 4 r 2 N A 4 r 2 N A 0 r r r or 2 2 r N A r r r N A r lim 0 r 0 r d 2 ss r N A 0 __________ (3) dr Integrating, 2 r N A constant __________ (4) 2 2 From equation (4), r N A r 0 N A 0 Substituting for N f Am equation (1), r 2C D d y AB A r 2 N A 1 y A d r 0 0 2 d r d y A r 0 N A 0 2 C D AB __________ (5) r 1 y A Boundary condition : At r = r 0 y A y AS And At r = y A y A Therefore equation (5) becomes, 2 1 y A r 0 N A 0 C D AB ln 1 y A y r r0 AS Simplifying, C D AB 1 y A N A ln __________ (6) 0 r 0 1 y A S Time required for complete evaporation of the droplet may be evaluated from making mass balance. Moles of water diffusing moles of water leaving the droplet unit time unit time 2 d 4 3 L 4 r 0 N A0 r 0 dt M A 2 L d r 0 4 r 0 __________ (7) M A d t Substituting for N from equation (6) in equation (7), A0 27 C D AB 1 y A L d r0 ln __________ (8) r0 1 y AS M A d t Initial condition : When t = 0 r 0= r 1 Integrating equation (8) with these initial condition, t L 1 1 0 d t r0 d r0 0 M A C D AB 1 y A r1 ln 1 y AS r 2 t L 1 1 M 2C D 1 y __________ (9) A AB ln A 1 y A S Equation (9) gives the total time t required for complete evaporation of spherical droplet of initial radius r 1 b) Combustion of a coal particle: The problem of combustion of spherical coal particle is similar to evaporation of a drop with the exception that chemical reaction (combustions) occurs at the surface of the particle. During combustion of coal, the reaction C + O 2 CO 2 cccurs. According to this reaction for every mole of oxygen that diffuses to the surface of coal (maximum of carbon), react with 1 mole of carbon, releases 1 mole of carbon dioxide, which must diffuse away from this surface. This is a case of equimolar counter diffusion of CO and 2. Norma 2y air (a mixture of N and 2 O 2s used for combustion, and in this case N does not 2es part in the reaction, and its flux is zero. i.e. N N 2 0 . The molar flux of O c 2ld be written as d y N CD O 2 y N N N O 2 O 2gas d r O 2 O 2 CO 2 N 2 __________ (1) Where D O 2gas is the diffusivity of 2 in the gas mixture. Since N N 2 0 , and from stoichiometry N O 2 N CO 2 , equation (1) becomes d yO N O C D O gas 2 __________ (2) 2 2 d r For steady state conditions, d r 2 N 0 __________ (3) d r O 2 Integrating, 28 r2 N O constant r02N O s __________ (4) 2 2 Where r is the radius of coal particle at any instant, and N O s is the flux of O 0 2 2 at the surface of the particle. Substituting for N O 2from equation (2) in equation (4), d yO 2 2 r 2CD O gas r N O s __________ (5) 2 d r 0 2 Boundary condition : r r0 yO y O s At 2 2 And At r yO 2 y O2 With these boundary condition, equation (5) becomes y 2 d r O 2 r0 NA 0 2 C D O 2gas d y O 2 r0 r yO s 2 which yields CD O2 gas N O 2 s yO 2 sy O 2 __________ (6) r 0 For fast reaction of O 2ith coal, the mole fraction of O 2 the surface of particle y O s 0 iz zero. (i.e.,) 2 . And also at some distance away from the surface of the particle y O 2 yO 2 0.21 (because air is a mixture of 21 mole % O an 279 mole % N 2) With these conditions, equation (6) becomes, 0.21 C D O 2 gas N O2 s ____________ (7) r0 Figure Combustion of a particle of coal 5. A sphere of naphthalene having a radius of 2mm is suspended in a large volume of shell air at 318 K and 1 atm. The surface pressure of the naphthalene can be assumed to be at 318 K is 0.555 mm Hg. The D ABof naphthalene in air –6 2 at 318 K is 6.92 * 10 /sec. Calculate the rate of evaporation of naphthalene from the surface. Calculation Steady state mass balance over a element of radius r and r + r leads to S N A r S N A r r 0 (1) 29 2 where S is the surface are (= 4 r ) dividing (1) by Sr, and taking the limit as r approaches zero, gives: d r N2 A 0 d r Integrating r N A= constant (or) 4 r N A= constant We can assume that there is a film of naphthalene – vapor / air film around naphthalene through which molecular diffusion occurs. Diffusion of naphthalene vapor across this film could be written as, N CD d y A y N N A AB d r A A B N B 0 (since air is assumed to be stagnant in the film) d y A N A CD AB y A N A d r y N CD d A A AB d r 1 y A dln 1 y A N A CD AB d r 2 W A Rate of evaporation = 4 r N A R = constant. 4 r 2CD d ln 1 y W AB A A d r W A d r 4 D AB C d ln 1 y A r 2 Boundary condition: 0.555 4 At r = R y A 7.303 * 10 760 –4 ln (1 – A) = 7.3 * 10 At r = y A= 0 ln (1 A) = 0 d r 0 Therefore W A 2 4 D AB C d ln 1 y A R r 7.3 *10 4 30 1 0 W A r 4 D AB C ln 1 y A 7.3 *104 R 1 4 W A 0 4 D AB C 0 7.3*10 R W A4 R D C * AB3 * 10 –4 5 C P 1.01325 * 10 Gas cons tant * T 8314 * 318 3 = 0.0383 kmol/m Initial rate of evaporation: Therefore W = A* 3.142 * 2 * 10 * 6.92 * 10 * 0.0383 * 7.3 * 10 –4 –12 = 4.863 * 10 –5 kmol/sec = 1.751 * 10 mol/hr. 3.5.5 Diffusion in Liquids: Equation derived for diffusion in gases equally applies to diffusion in liquids with some modifications. Mole fraction in liquid phases is normally written as ‘x’ (in gases as y). The concentration term ‘C’ is replaced by average molar density, . M av a) For steady – state diffusion of A through non diffusivity B: N A constant , N = B D AB N A xA1 x A2 z x BM M av where Z = Z – 2, th 1length of diffusion path; and X B2 X B1 X BM X B2 ln X B1 b) For steady – state equimolar counter diffusion : N A N = Bonst D AB D AB N A C A1 C A2 x A1 x A 2 Z Z M av 4. Calculate the rate of diffusion of butanol at 20C under unidirectional steady state conditions through Are you sure you want to buy this material for You're already Subscribed! Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

Vibroacoustical Analysis of Multiple-Layered Structures with Viscoelastic Damping Cores This paper presents a modeling technique to study the vibroacoustics of multiple-layered viscoelastic laminated beams using the Biot damping model. In this work, a complete simulation procedure for studying the structural acoustics of the system using a hybrid numerical model is presented. The boundary element method (BEM) was used to model the acoustical cavity, whereas the finite element method (FEM) was the basis for vibration analysis of the multiple-layered beam structure. Through the proposed procedure, the analysis can easily be extended to another complex geometry with arbitrary boundary conditions. The nonlinear behavior of viscoelastic damping materials was represented by the Biot damping model taking into account the effects of frequency, temperature, and different damping materials for individual layers. The curve-fitting procedure used to obtain the Biot constants for different damping materials for each temperature is explained. The results from structural vibration analysis for selected beams agree with published closed-form results, and results for the radiated noise for a sample beam structure obtained using a commercial BEM software are compared with the acoustical results of the same beam by using the Biot damping model. The traditional designs of free-layer, constrained-layer or sandwich-layer, damping treatment using viscoelastic materials have been around for over forty years. Recent improvements in the understanding and application of the damping principles, together with advances in materials science and manufacturing, have led to many successful applications and the development of patch damping and multiple-layered damping structures. The key point in any design is to recognize that the damping material must be applied in such a way that it is significantly strained whenever the structure is deformed in the vibration mode under investigation. Numerous researchers have successfully implemented the passive constrained layer (PCL) and active constrained layer (ACL) systems. In 1959, Kerwin and Ross et al. presented a general analysis of viscoelastic material structure. The damping was attributed to the extension and shear deformations of the viscoelastic layers. Ditaranto developed sixth-order equations of motion in terms of axial displacements and developed a closed-form solution. Mead and Markus extended the sixth-order equations of motion for transverse displacement to include various boundary conditions. A paper by Rao presented the equations of motion of viscoelastic sandwich beams with various boundary conditions using the energy method. The equations were solved numerically, and a practical design guideline was presented. Similar to Rao’s theory, Cottle used Hamilton’s principle to derive equations of motion. The damping could also be increased by adding passive stand-off layer (PSOL) and slotted stand-off layer (SSOL) to the layered systems. Falugi and Parin et al. conducted theoretical and experimental work on a four-layered panel and a five-layered beam with PSOL treatment. Rogers and Parin and Yellin et al. have performed experimental investigations and demonstrated that PSOL treatment increased the damping significantly in aeronautical structures and beams. Yellin and colleagues [11, 12] also developed normalized equations of motion for beam, fully treated with PSOL using nonideal stand-off layer assumption. The equations were solved using the method of distributed transfer functions . In addition to the closed-form analytical approach, many researchers have used the finite element method (FEM), the most popular numerical modeling method in building the numerical model of the multiple layers system. In 2000, Chen and Chan studied four different types of integral FEM models with the viscoelastic cores. The numerical stability and accuracy as well as for convergence issue of these four different FEM models were demonstrated by comparing the numerical results with those from experiments. Lesieutre and Lee proposed a 3-node, 10 DOF FEM model for the three-layer ACL damping beam. This FEM model is advantageous in active control application due to its features of nonshear locking and adaptability to segmented constraining layers. Other than the Hamilton and FEM numerical methods in building the models, other researchers have proposed many irregular modeling techniques for the numerical representation of continuous/discontinuous systems. Kung and Singh calculated the natural frequencies and loss factor using the Rayleigh-Ritz energy method and modal-strain energy technique in modeling a 3-layer patch damping structure. These approximate modeling methods were also extended to rectangular damping patch of plates and shells with viscoelastic cores. Zhang and Sainsbury combined the Gerlerkin orthogonal function with the traditional finite element method and successfully applied to the vibration analysis of the damped sandwich plates. While the FEM is used widely in the modeling of the structure, many researchers sought for proper mathematical models to represent the damping behavior of the viscoelastic material, as well as incorporating the damping model in commercial FEM software packages. Currently, many FEM commercial software incorporate damping models based mostly on viscous/hysteretic damping. Some allow incorporation of damping energy dissipation in the time domain using the Prony series. None of these damping models, however, is suitable to capture the damping behavior in the frequency domain, which is the most important issue in predicting the vibro-acoustical response of complex structural systems. The drawback of these damping models raised considerable interest and motivation in the development of damping models of viscoelastic material in the frequency domain compatible with FEM software. These damping models can be classified as derivative type and integral type. The “Fractional Derivative” is essentially the representative damping model in the derivative form family proposed by Bagley and Torvik in 1983. This damping model not only described the material properties of viscoelastic damping but so established the closed-form equation compatible with the FEM technique. Compared with the other integral-form models, the fractional derivative is only able to capture the relatively weak frequency-dependent information; however, it was an important milestone in the area of damping research. Lesieutre et al. mathematically modeled the relaxation behavior of viscoelastic material in terms of augmenting thermodynamic field (ATF) in 1989. Initially, introducing a single augment field, this damping model provided the ability to represent the light-damping behavior, with the application of a 1D viscoelastic structure. In the subsequent research, using a series of augment fields, the ATF model is able to model the damping material of higher loss factor with the weak frequency dependence. Remedying the limitation of 1D application, Lesieutre and Lee proposed an anelastic displacement field (ADF) technique in 1996 and successfully extended its application from the 1-D problem to the 3-layer sandwich beam and 3-D problems. As far as the mini-oscillator damping models are concerned, the complex shear modulus which is a function of both frequency and temperature can be expressed by a series of mini-oscillation perturbations. Biot first proposed the first-order relaxation function with the introduction of the “dissipative variables” into the dynamic equations using the theory of irreversible thermodynamics. In 2007, Zhang and Zheng utilized the Biot model to describe the dynamic behavior of a viscoelastic structure. The dimension reduction technique and nonlinear curve-fitting procedure were discussed in the paper. McTavish developed another mini-oscillator damping model called “GHM” by the usage of Second-order relaxation function. Compared with the Biot model, the GHM model has a more complicated expression and also requires better performance of the computational tool. The popularity of these integral-form damping models in recent years brought two research interests: nonlinear curve fitting and dimension reduction. The advanced curve-fitting techniques in the damping models guarantee the accuracy of the numerical representation of the actual shear modulus data from the experiment. The dimension reduction technique increases the computational efficiency due to the additional orders of equation in order to gain the frequency independence of the frequency-form damping model. Zhang et al. converted the nonlinear curve-fitting problem in frequency domain with respect to the GHM parameters into the constrained nonlinear optimization problem. The efficiency and correctness were demonstrated for a commercial viscoelastic material. Park et al. examined the GHM damping model with the application to the FEM method associated with the Guyan reduction technique. The numerical example in this research leads to an FEM model applied to the GHM dynamic equation quantitatively without increasing the number of order. Hao and Rao carried out the optimum design of a three-layer sandwich beam for the vibration analysis in 2005. In this research, the numerical model is a comprehensive formulation for a three-layer unsymmetrical sandwich beam with two different damping materials adjacent to each other. The criterion of the optimization is to minimize the mass of the structure while maximizing the system damping. In 2008, Lee published the semicoupled vibroacoustical analysis and optimization of a simply supported three-layer sandwich beam. The modal superposition method was used to investigate the vibration problem with the fractional derivative damping model. The interior acoustical problem was studied by BEM numerical technique, and the optimization problem was established through the appropriate sizing parameters of the sandwich beam. The objective of this paper is to extend the previous work by the authors on the vibration analysis of a multiple-layered beam structure incorporating the Biot damping model to solve the acoustic problem to predict the radiated noise. In this paper, we present a complete numerical procedure for the vibroacoustical analysis and design for a multiple-layer laminated damping beam. Results obtained from the proposed vibration analysis are compared with the previous closed-form results to show the validity of this approach. The radiated noise spectrum at selected field point shows good agreement between the 2-D BEM acoustical analysis and the result without system damping calculated by commercial software for a sample viscoelastic damping structure. The acoustical solution is demonstrated, and the correlation between sound pressure level (SPL) and the loss factor is also highlighted. 2. FEM Modeling and the Biot Dynamic Equation The FEM modeling procedure and the establishment of the Biot dynamic equation will be discussed in this section. The structure chosen for illustration is a seven-layer viscoelastic sandwich beam. The elastic beam and the constrained damping layer are the two fundamental components in this FEM-modeling technique. The concept of transfer matrix is used to convert the local coordinates to the global coordinates in order to assemble and construct the complete model of the sandwich damping structure with arbitrary number of layers. The Biot viscoelastic damping model will be used to describe the damping behavior. Through the use of the FEM, the structure is discretized which will enable the use of the Biot damping model for different damping layers in the structure. The reader is referred to the nomenclature for the definition of different variables used in the derivation. 2.1. FEM Modeling of Component I: The Elastic Layer Figure 1 shows the elastic layer in the FEM model, containing 2 nodes and 6 degrees of freedom (DOF). The element displacements of each node can be expressed as follows: The stiffness matrix can be derived based on the following energy method: as the shape functions are the following: in which : the local coordinate, , , : longitudinal length of elastic layer, : cross-sectional area of the elastic layer, : Young’s modulus of the elastic layer, and : moment of inertia of elastic layer. Similarly, the element mass matrix can be expressed as: 2.2. FEM Modeling of Fundamental Component II: The Constrained Damping Layer The FEM model of the constrained layout containing the damping layer sandwiched between two outer layers is shown in Figure 2. This Figure illustrates each element consisting of 2 nodes and 8 DOF, where the nodal displacement vector is as follows: Through the introduction of transfer matrix, in which each means the following vector: ; the element elastic stiffness and the element viscoelastic stiffness matrix for this 3-layer component, respectively, are the following: where : cross-sectional area of the damping layer, : long-term shear modulus of the damping layer, and : correction factor of the shear strain energy, for the rectangular cross-section, . Also, the element mass matrix for this 3-layer component is where 2.3. FEM Modeling of a Seven-Layer Constrained Damping Beam The seven-layer sandwich beam consists of seven alternating layers—four elastic layers and three damping layers. Figure 3 shows the FEM model of a seven-layer sandwich beam containing 2 nodes and 10 DOF, and the node displacement vector is as follows: The transfer matrix to obtain the element stiffness and the mass matrix when the 1st, 3rd, 5th, and 7th layers are elastic are follows: Similarly, the element stiffness and the mass matrix for the 2nd, 4th, and 6th layers of the constrained damping layer can be derived through the transfer matrix: where the notation means Based on the above equations and design parameters of each layer, the element mass/stiffness/damping matrix of the seven-layer sandwich damping beam can be expressed as follows: Thus, the element matrices can be assembled to obtain the global mass/stiffness/damping matrix and can be applied to the boundary condition through the conventional FEM technique. Taking into the consideration of the viscoelastic damping properties, the global matrices need to be manipulated as a portion of the Biot dynamic equation. 2.4. Introduction of the Biot Dynamic Equation To consider the vibration problem numerically, the dynamic equation discretized by FEM technique needs to be expressed by the following second-order ordinary differential equation (ODE) form: The Biot viscoelastic damping model numerically represents the complex shear modulus with a series of mini-oscillator perturbing terms: in which is the long-term shear moduli; and are the Biot constants. These parameters are positive and can be determined by nonlinear curve fitting from the experimental data. The curve-fitting procedure will be discussed in Section 3. Substituting the Biot damping model into (15), the dynamic equation with terms of the Biot parameters for the first viscoelastic material and terms for the second viscoelastic material can be developed as follows: where and are the eigenvector and diagonal eigenvalue matrices, respectively, from the damping matrix . Additionally, , , and denote terms of the Biot parameters and the dissipative coordinates, respectively, for first viscoelastic material. Similarly, , , and denote terms of the Biot parameters and the dissipation coordinates, respectively, for second viscoelastic material. A detailed derivation can be found in the previous publication . 3. Parametric Determination of the Biot Damping Model A curve-fitting technique is used to provide the accurate Biot constants to the dynamic equation and to establish the dynamic characteristics of the viscoelastic materials. In this section, the nonlinear curve-fitting procedure for the complex shear modulus in the frequency domain is converted into a nonlinear constrained optimization problem. The complex shear modulus with the Biot damping model form can be broken into real and imaginary parts separately: The Biot parameters—, , and —are estimated from experimental data with the certain fitting frequency range, on real part and imaginary parts separately. Generally speaking, one set of the Biot parameters needs to be determined for each ambient temperature independently. In (18), is the number of the Biot perturbing items, defining the capability of this numerical approximation. As the Biot terms () are increased, the relative error between the experimental data and the curve-fitting result reduces. Assuming ; ; ; ; ; with the constraint condition: ; , the target equation of the optimization problem is the following: In the target equation (19), stands for the complex shear modulus from the experimental data with interested points (larger than the number of unknowns). The 3 M ISD-110/112 viscoelastic polymer is selected in this simulation. The experimental data is obtained by the Arrhenius empirical equation from . With a specific fitting range at a particular temperature, the complex shear modulus can be synthesized from one set of the Arrhenius coefficients. The number of terms () in (18) needs to be determined to ensure the precision of this approximation. The curve fitting of the experimental data is accomplished using the commercial software package Auto2fit on the real and imaginary parts simultaneously. Using the Biot terms equal to six and four with respect to two commercial damping materials 3 M ISD-110 and 112, respectively, the results are shown in Table 1 for ambient temperature () equal to 45°C and frequency range of 500 Hz. As shown in Figures 4(a) and 4(b), the Biot parametric determination technique estimates the dynamic properties of 3 M ISD-110/112 at 45°C with almost zero error. The constants determined using the above procedure along with the FEM model of sandwich beam will now be incorporated to solve the complete Biot dynamic equation using the decoupling transformation technique. 4. Decoupling Transformation and Dynamic Solution In this section, the algorithm used to obtain the frequency response function (FRF) will be discussed with respect to the vibroacoustical problem for a multiple-layer viscoelastic damping structure. In this research, the damping matrix in (15) does not have a proportional relationship with the mass and stiffness matrix. Thus, a decoupling transformation is needed to construct the first-order state equation by introducing the auxiliary equation as follows: where Here, is the number of DOF in the , , and matrices, the DOF of and matrices is . Firstly, the free vibration of (20) will be considered. Assuming , the following form of solution is obtained: or where matrix stands for complex conjugate eigenvalues including the natural frequencies and loss factors information: It must be noted that zero items will appear in the eigenvalue matrix if the damping matrix is not fully ranked. The mode shape vector for the vector can be extracted from the eigenvector matrix with respect to the vector : In addition, (22b) can be numerically solved by using mathematical software package such as MATLAB or Mathematica. Secondly, the forced vibration solution of (20) in the time domain will be discussed. Assuming , the variable substitution can be made by assuming , converting the state-space equation from the time space to the modal space. By left multiplying of with the substitution of , we get: The diagonal modal mass and stiffness matrix are: Then rewrite the equation with the diagonal mass and stiffness matrices The FRF in the frequency domain can be easily determined through the complex conjugate eigenvalue matrix , eigenvector matrix , and the modal mass matrix . The modal scaling factor matrix can be calculated through the following: Thus, FRF can be established through the modal parameters, being expressed in partial fraction form in terms of the residue vector and system poles as follows: The system velocity can now be obtained from the above equation by a simple Fourier transformation. By doing so, the vibration problem can be extended to an FRF-based acoustical problem and the combination of these two analyses is the particle velocities information calculated by the following: 5. Acoustical Boundary Element Method (BEM) Analysis 5.1. Introduction of Acoustical BEM Theory In Section 4, the vibration problem of the multiple-layer sandwich beam is solved through the time-domain dynamic ordinary differential equation of the Biot damping model with numerical analysis by the FEM technique. The vibration problem can be extended to the acoustical problem by the semicoupled method: the vibration will induce a change in sound pressure, yet the sound pressure will not cause the vibration. In this section, the acoustical interior problem will be numerically solved by 2D boundary element method (BEM) technique in a bounded fluid domain as shown in Figure 6. The sound pressure distribution () of the time-harmonic wave in the domain satisfies the governing differential equation, well known as the Helmholtz equation, associated with the boundary conditions on boundary as follows: Here, is equal to , which means that the wave number is equal to the radiant frequency over the speed of sound; , , stand for the normal velocity, density of the fluid (normally the air), and acoustical impedance of the fluid , respectively. In this work, the link between the vibration and the acoustics analysis is the normal velocity at the acoustical boundaries. Recalling the dynamic solution of the decoupling transformation, the particle velocity in the time domain at each node can be calculated through (30) if the multiple-layer sandwich beam is discretized by the FEM; alternatively the FRF, the complex ratio between the output and input response in the frequency domain, can be determined through (29). Once the input signal is given, the particle velocity of the system displacement versus frequency relationship can be conveniently obtained through the FRF. To solve the governing differential equation (31) in the bounded fluid domain , the Helmholtz Equation can be transformed into the integral equation, converting the 2-D area integration to the 1-D curve integration around the area: in which : geometry-dependent coefficient, normally when is in the domain and when is on the smooth boundary , : sound pressure at source point , : is the field point and for the 2D BEM problem the Euclidian distance between and , and the Second-type Henkel function, : normal vector pointing away to the fluid domain . By discretizing the boundary into a series of curve-linear elements through the introduction of the shape functions, the integral equation can be calculated numerically by solving the following linear matrix: where comes from the terms of and , is derived from , and the vector and include sound pressure and particle velocity values, both unknowns and known from the boundary condition. Thus, each set of node velocities due to the force input results in one set of solutions on the sound pressure by BEM discussed in this section. In sum, through the proposed acoustical BEM, it is possible to compute the time-harmonic sound pressure distribution corresponding to each single frequency point in the frequency spectrum. 5.2. Calculation Details in This BEM Analysis For this particular acoustical BEM interior problem, the boundary of acoustical cavity is discretized as 18 quadratic equally spaced boundary elements. The quadratic curvilinear element has three nodes, and the interpolation between each node represents the geometry of each element. The shape functions are as folows: with respect to the following element coordinates: where and are the coordinates at each nodal point, and stands for the local coordinate between −1 and 1 on a master element. When the seven-layered sandwich beam ( m) is simply supported at the bottom of the acoustical cavity, the sound pressure level at the field point ( m, m) is calculated through this proposed method, and the calculation results are presented in Section 6. Figure 7 demonstrates the detailed layout of this 2D acoustical cavity problem. The anechoic boundary condition is applied on the inside of the acoustical cavity, and the thickness of the multiple-layered beam is neglected. 6. Numerical Results and Discussion 6.1. Design Parameter of Sandwich Beam and Vibration Analysis Result The data presented in Table 2 are used to predict the vibration performance of the system using the numerical simulation method presented in this paper, and the results are compared with the closed-form solution of Hao . The curve-fitting results for the damping material 3M ISD-110 at 45°C discussed earlier are selected for the shear modulus of the viscoelastic layers in this example. The results are shown in Table 3. It shows that the simulation presented in this paper conforms to the closed-form solution. This validates the analysis methodology proposed in the paper. 6.2. Frequency-Spectrum Analysis under the Arbitrary Input Figure 9 shows the transverse velocity of the middle node (node number 7) with a 10 N step input in the frequency domain vertically applied at the middle (node number 7) of the simply-supported seven-layer sandwich beam with the same design parameters as the previous example. The same curve-fitting results of 3 M ISD110 at the ambient temperature of 45°C for the shear modulus are used in this example. This pivotal result is the demonstration of extending the vibration to the acoustical problem in the frequency domain when an arbitrary force is applied on the structure. 6.3. Acoustical BEM Results Figure 10 illustrates the contour plot ( Hz) of SPL when the seven-layer sandwich beam (using the same design parameters as before) is subjected to a 10 N step input in the frequency domain at the middle node. The interpolation of each elements result in Figures 10 and 11 shows the continuous sound pressure distribution in the acoustical cavity with an anechoic boundary condition. Figure 12 extracts the frequency spectrum of SPL at the filed point (0.5, 0.4 m) indicated by red dot in Figure 10. From the results of Figure 12, it can be found that the dominant contribution is due to the peak value of the first flexible vibration mode, which is in agreement with the frequency-spectrum analysis of the vibration problem. 6.4. Validation Using a BEM Commercial Software In this section, a hybrid FEM-BEM model of a beam without the viscoelastic damping was developed using the commercial software packages ANSYS ADPL and LMS Virtual. Lab Acoustics. The harmonic vibration analysis is conducted in ANSYS APDL module, and the frequency spectrum of field point SPL was calculated in Virtual.Lab Acoustics module for comparison with the SPL frequency spectrum presented in Section 5. The analysis sequence consists of the following steps.(a)Build the FEM model and apply appropriate boundary conditions in ANSYS ADPL. The 8-node element SOLID45 (element size =10 mm for each layer) was used to build the 3D seven-layer model. The design parameters are identical with the parameters in Tables 1 and 2 for the comparison and the geometry boundary conditions are simply supported. A 10 N force at each frequency is applied at the middle nodes.(b)Conduct the harmonic vibration analysis in ANSYS ADPL. The harmonic analysis is used to calculate the nodal displacements for a forced vibration problem in the frequency domain. The frequency range is 0–200 Hz with a 2 Hz for step size, and the full method is being utilized in this analysis. The comparison of system frequencies between ANSYS modal results and calculation results by the Biot dynamic equation is shown in Table 4. The results show that the 3D model built in ANSYS APDL has good correlation with the FEM model. (c)Prepare the BEM mesh in LMS Virtual. Lab Pre-Acoustics module. It converts from a solid FEM model to a skin mesh that the BEM analysis requires. The BEM mesh, can be seen as a wrap around the structural mesh and usually the BEM mesh is coarser.(d)Calculate the sound pressure in LMS Virtual.lab Acoustics module. Both acoustical and structural meshes are imported to VL Acoustics. The nodal displacement at each vibration mode calculated in ANSYS APDL is also imported and mesh-mapped to the acoustical skin mesh as the vibration boundary condition. The location of field plane and field point is consistent with the 2D BEM analysis in this research. The acoustical pressure is solved over the frequency range from 2 to 200 Hz. As shown in Figure 13, the peak frequency from the 2-D BEM calculation matches with the first dominant SPL peak obtained from the VL Acoustic result without the damping. Comparing the two results, it is clear that the introduction of viscoelastic damping not only causes almost a 20 dB reduction in the first peak SPL but also attenuates the sound at other peaks as well. This proves that the use of viscoelastic damping material will greatly attenuate the vibroacoustical response of the structure. 6.5. Acoustical Performance for a Combination of Several Viscoelastic Materials at Different Temperatures The temperature is a significant external factor affecting the performance of viscoelastic damping material in a mechanical system. With an increase in temperature, the loss factor approaches its best performance towards the transition region and then decreases afterwards. In this example, the objective is to study the effects of both 3 M ISD110 material (that has a better damping performance) and the 3 M ISD112 over the chosen temperature between 40 and 60 degree Celsius. It is of interest to study the effect of the combination of these two materials on the damping of the structure. To introduce the different viscoelastic materials, the seven-layer sandwich beam (with the same parameters as in the previous example) is redesigned incorporating both damping materials (3 M ISD110 and ISD112). This system is compared to an identical structure with only one damping material (either 3 M ISD110 or ISD112). In the system including two viscoelastic materials, the outer damping layers (2nd and 6th) are 3 M ISD112 and the inner damping layer (5th) is the 3 M ISD110. The simply supported boundary condition is examined in this numerical example, and the temperature range is from 40 to 60 degree Celsius. The acoustical response is also calculated with the step input in the frequency domain (equivalent to impulse input in the time domain). Table 5 shows the first order natural frequency, the system loss factor, and the corresponding peak value (dB) of the sound pressure level over the temperature range with the simply supported boundary condition applied to the FEM model. It can be seen that for the same damping material, as the ambient temperature is increased, the value of SPL increases while the loss factor decreases. A framework for conducting vibro-acoustical analysis for multiple-layer beam structures containing different types of viscoelastic materials is presented in this paper. Several observations and conclusions can be drawn from the results of this research.(1)The vibration section of the proposed analysis consists of FEM model of multiple-layered damping beam incorporating the Biot damping model. The FEM model of the beam structure can be extended to more complicated damping structures using the same procedure. The nonlinear curve-fitting technique accurately estimates the Biot constants. The Biot damping model can then be solved using the decoupling transformation to yield the frequency-spectrum analysis.(2)The Biot damping model is also capable of improving a structure’s damping performance by adding new features such as different viscoelastic materials and the variation of operating temperature. The result obtained through the procedure of vibration analysis discussed in this paper compares well to the closed-form solution from a previous work. The first peak from the frequency spectrum is the predominant cause of the vibration issue in this damping structure.(3)The direct boundary element method of analysis for acoustical cavity applied under anechoic boundaries was chosen as the basis for predicting the particle velocity from the frequency-spectrum analysis. The acoustical result validates the frequency-spectrum result from vibration analysis and has good agreement with the predicted SPL spectrum of the identical sandwich beam without damping calculated by commercial software. |, :||Elastic stiffness/viscous stiffness matrix| |, :||Coefficient matrix of state equation| |:||Dissipation coordinate vector| |, :||Number of mini-oscillators for first/second type of viscoelastic material| |, :||FEM shape function of longitudinal/transverse deflection| |:||Number of DOF| |:||Density of material| |:||Thickness of layer| |:||Length of beam| |, :||Biot constants| |:||Nodal normal component of boundary velocity.| The authors (D. Rao and F. Lin) hereby declare that they do not have any direct or indirect financial relation leading to any conflict of interests with the commercial identities (BEM software, FEM software, Auto2fit, MATLAB, and Mathematic) mentioned in the text of their paper. E. M. Kerwin, “Damping of flexural waves by a constrained visco-elastic layer,” Journal of the Acoustical Society of America, vol. 31, pp. 952–962, 1959.View at: Google Scholar D. Ross, E. E. Ungar, and E. M. Kerwin, “Damping of plate flexural vibration by means of viscoelastic laminae,” in Structural Damping-a Colloquium on Structural Damping Held at the ASME Annual Meeting, pp. 49–87, 1959.View at: Google Scholar R. A. Ditaranto, “Thery of vibratory bending for elastic and viscoelastic layered finite-length beams,” Journal of Applied Mechanics, vol. 32, pp. 881–886, 1965.View at: Google Scholar D. J. Mead and S. Markus, “The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions,” Journal of Sound and Vibration, vol. 10, no. 2, pp. 163–175, 1969.View at: Google Scholar D. K. Rao, “Frequency and loss factors of sandwich beams under various boundary conditions,” Journal of Mechanical Engineering Science, vol. 20, no. 2, pp. 271–282, 1978.View at: Google Scholar E. T. Cottle, Damping of layered beams with mixed boundary conditions [M.S. thesis], Air Force Institute of Technology, 1990. M. Falugi, “Analysis of a five-layer, viscoelastic, constrained-layer beam,” in Proceedings of the Damping Workshop Paper CCB, 1991.View at: Google Scholar M. Parin, L. C. Rogers, and M. Falugi, “Practical stand off damping treatment for sheet metal,” in Proceedings of the Damping Workshop Paper IBA, 1989.View at: Google Scholar L. C. Rogers and M. Parin, “Experimental results for stand-off passive vibration damping treatment,” in Proceedings of the Smart Structures and Materials 1995: Passive Damping, pp. 374–383, March 1995.View at: Google Scholar J. M. Yellin, I. Y. Shen, P. G. Reinhall, and P. Y. H. Huang, “Experimental investigation of a passive stand-off layer damping treatment applied to an Euler-Bernoulli beam,” in Proceedings of the 1999 Smart Structures and Materials—Passive Damping and Isolation, vol. 3672 of Proceedings of SPIE, pp. 228–233, March 1999.View at: Google Scholar J. M. Yellin, I. Y. Shen, P. G. Reinhall, and P. Y. H. Huang, “An analytical and experimental analysis for a one-dimensional passive stand-off layer damping treatment,” Journal of Vibration and Acoustics, vol. 122, no. 4, pp. 440–447, 2000.View at: Google Scholar B. Yang and C. A. Tan, “The transfer functions of one dimensional distributed parameter systems,” Journal of Applied Mechanics, vol. 116, pp. 341–349, 1959.View at: Google Scholar S. W. Kung and R. Singh, “Vibration analysis of beams with multiple constrained layer damping patches,” Journal of Sound and Vibration, vol. 212, no. 5, pp. 781–805, 1998.View at: Google Scholar R. L. Bagley and P. J. Torvik, “Fractional calculus-a different approach to the analysis of viscoelastically damped structures,” AIAA Journal, vol. 21, no. 5, pp. 741–748, 1983.View at: Google Scholar M. Hao, Vibration analysis of constrained layered beams with multiple damping layers [Ph.D. thesis], Michigan Tech, 2005. G. A. Lesieutre, E. Bianchini, and A. Maiani, “Finite element modeling of one-dimensional viscoelastic structures using anelastic displacement fields,” Journal of Guidance, Control, and Dynamics, vol. 19, no. 3, pp. 520–527, 1996.View at: Google Scholar D. J. McTavish, “Shock response of a damped linear structure using GHM finite elements,” in Proceedings of the 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, pp. 1681–1689, April 2003, Paper 1591.View at: Google Scholar L. Zhang, H. P. Du, Y. M. Shi, and X. Z. 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3 edition of Mathematics, creation and study of form found in the catalog. Mathematics, creation and study of form Jacqueline Pascal Evans |Statement||by Jacqueline P. Evans.| |Series||Addison-Wesley series in introductory mathematics| |The Physical Object| |Number of Pages||358| Countless math books are published each year, however only a tiny percentage of these titles are destined to become the kind of classics that are loved the world over by students and mathematicians. Within this page, you’ll find an extensive list of math books that have sincerely earned the reputation that precedes them. For many of the most important branches of mathematics, we’ve. Form 4 from Text Book Centre. Books, Stationery, Computers, Laptops and more. Buy online and get free delivery on orders above Ksh. 2, Much more than a bookshop. Mathematics is the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. Mathematics is one of the great gifts of God and greatest discoveries of the human race. Mathematics 10 Academic/Principle of Mathematics 10 (MPM2D) Paperback – Jan. 1 This book cannot be relied for independent study. There are gaps left which require student consult a textbook to understand the subject. The guide itself does not guide to work oh his own. I don't understand why it is priced so high.4/5(7). Principles of Mathematics Book 2 lays a solid foundation—both academically and spiritually—as your student prepares for high school algebra! Students will study pre-algebra concepts, further develop their problem-solving skills, see how algebraic concepts are applied in a practical way to everyday life, and strengthen their faith!%(5). 1. Introduction. The late fifth and fourth centuries B.C.E. saw many important developments in Greek mathematics, including the organization of basic treatises or elements and developments in conceptions of proof, number theory, proportion theory, sophisticated uses of constructions (including spherical spirals and conic sections), and the application of geometry and arithmetic in the. Space station crew safety alternatives study, final report. Cats, big and little How to wow Aseptic processing of foods containing solid particulates Proverbs for the people Guidelines for preparing and filing drug submissions law of torts Dessous Postcard Book (PostcardBooks) interpretation of vapor-phase infrared spectra Influence of flow conditions on the backscattered doppler ultrasound signal. Beginning structured Cobol Lisp based CIRCAL environment. Additional Physical Format: Online version: Evans, Jacqueline P., Mathematics: creation and study of form. Reading, Mass., Addison-Wesley Pub. Happily, the book is lucidly written and neither subset of students (mathematics or physics) should face any difficult in perusing the exposition. Its efficacy for mathematics students accepted, I concentrate upon my reasons for suggesting this unique exposition for the later category of Cited by: Happily, the book is lucidly written and neither subset of students (mathematics or physics) should face any difficult in perusing the exposition. Its efficacy for mathematics students accepted, I concentrate upon my reasons for suggesting this unique exposition for the later category of 5/5(11). Mathematics is Discovery and not Invention. This all means that mathematicians aren’t making up what they do. It’s not a case of someone with a blank sheet of paper asking what they’d like mathematics to be. Mathematics is already been there and has always been – part of the eternal and beautiful order of the mind of God. they need to learn Mathematics. It also means that the book helps Form I students to recall their mathematical knowledge from primary school and translate this into English. • Tanzanian. Mathematics is used in Tanzania on a daily basis. Mathematics was developed by men and women from different parts of the world in response to human needs, to. MATHEMATICAL CREATION is an article from The Monist, Volume View more articles from The this article on this article's JSTOR. Mathematics books Need help in math. Delve into mathematical models and concepts, limit value or engineering mathematics and find the answers to all your questions. It doesn't need to be that difficult. Our math books are for all study levels. Others require spectroscopic study to detect each star’s particular identity or fingerprint. The Bible states in I Corinthians that “star differs from star in splendor.” How could ancient man have known this. It is only logical to conclude that the Creator of the stars chose to tell man to include this in the Bible. Examples: Decimals on the Number Line Example 5 a) Plot on the number line with a black dot. b) Plot with a green dot. Solution: For we split the segment from 0 to 1 on the number line into ten equal pieces between 0 and 1 and then countFile Size: KB. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in pure mathematics. Statistics and other decision sciences. How to Create a Basic Study Guide. Study guides can provide you with a quick and easy way to go over important material before tests or exams. There are a number of different basic study guide formats and each is designed to help you 82%(17). The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and mo Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. Principles of Mathematics Book 1 goes beyond adding a Bible verse or story to math instruction, it actively teaches and describes how the consistencies and creativity we see in mathematical concepts proclaim the faithful consistency of God Himself and points students towards understanding math through a Biblical worldview%(22). Mathematics Test Practice Book This practice book contains one actual, full-length GRE® Mathematics Test test-taking strategies Become familiar with test structure and content test instructions and answering procedures Compare your practice test results with the performance of those who took the test at a GRE Size: 1MB. MATHEMATICS: HOME FORM 1 FORM 2 MID-YEARLY EXAMINATIONS FORM 1 MID-YEARLY EXAMINATIONS FORM 2 ANNUAL EXAMINATIONS FORM 1 ANNUAL EXAMINATIONS FORM 2 Interactive Resources F1 Interactive Resources F2 Form 2 Booklets Powered by Create your own unique website with customizable templates. On this page you can read or download Download Zimsec O Level Mathematics Green Books in PDF format. On this page you can read or download download zimsec o level mathematics green books in PDF format. lo p m e n t C o., L td. (F ro n t C o ve r Im ag e.). A rc h ite cts. (P h o to. R ig h t). 18 PROJECT CASE STUDY Envisioning. About this Item: Cambridge University Press. Paperback. Condition: new. BRAND NEW, Mastering Mathematics Form 3 Student's Book: Level 3, Tamambang Andrew Tangang, Akah Andrew Mua, Napthalin Achubang Atanga, Ashu Frida Tikunyen, Njoh Joseph Molombe, Mastering Mathematics Form 3, the third book in an exciting new series for Ordinary Level Mathematics, is specially developed for. When I first encounter the vast topic REAL ANALYSIS, searched internet for the best books available on this topic But I never found books that explains me like Iam a child (Just kidding right!!!) Well I got the best book in my hand which is “ELEM. + Free Mathematics Ebooks by has a list of online mathematics books, textbooks, monographs, lecture notes, and other mathematics related documents freely available on the web, arranged alphabetically. I didn’t go through all of the list but a majority of the ebooks are either in HTML or PDF formats. Mathematics Quotations: “Philosophy is written in this grand book -- I mean the universe -- which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of.Creation Book Publishers Other helpful sites Creation Ministries International (CMI) exists to support the effective proclamation of the Gospel by providing credible answers that affirm the reliability of the Bible, in particular its Genesis history.Mathematics is on the artistic side a creation of new rhythms, orders, designs, harmonies, and on the knowledge side, is a systematic study of various rhythms, orders.– William L.

Today’s IFOD is about “The Golden Ratio” otherwise known as “phi.” The golden ratio is pretty interesting, but first, let’s discuss reproduction of bunnies (or you can just skip down to the meat of the golden ratio below): In 1202 Leonardo de Pisa, popularly known as Fibonacci, posed the following question: “A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?” If you assume that each pair consists of one male and one female and that they are able to procreate after their first month and there is a one-month gestation period and that the bunnies are monogamous, here’s what you get: - You begin with a pair of baby rabbits at the beginning of the first month - A month later that pair of bunnies is now adult and the following month they have their first pair of bunnies. - So, after the first month you have one pair. After two months you have one pair. After three months there are two pair. In the fourth month the original pair gives birth again, while the new pair from month three grows to adulthood. So, there are three pair. Following still? In the fifth month the original couple produces yet another pair, and now the first set of offspring produce offspring. Now there are five pair. The next month there are eight pair. And so on. This sequence of numbers is referred to as the Fibonacci Sequence. Not because this is necessarily how rabbits reproduce, but because of the importance of the sequence that the rabbit example produces: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 . . . . Ignoring the rabbit aspect of the example above, you create the Fibonacci sequence by adding together the first two numbers to get the third, and then the second and third to get the fourth, and so on. Now for the Golden Ratio: If you take the ratio of consecutive Fibonacci numbers, the resulting ratio settles down into what is known as “the Golden Ratio”, or 1.6180339887498948482… Example: - 1/1 = 1 - 2/1 = 2 - 3/2 = 1.5 - 5/3 = 1.6666… - 8/5 = 1.6 - 13/8 = 1.625 - 21/13 = 1.61538… - 34/21 = 1.61904… - 55/34 = 1.61764… - 89/55 = 1.61818… - And so on . . . We find the golden ratio when a line is segmented so that the whole length of the line divided by the long part is also equal to the long part divided by the short part. A Golden Rectangle is one where the ratio between the length and height is Phi (so, if a = 1, a+b = 1.618. . . ). The Golden Ratio appears all throughout nature, in art, architecture, music and more. Often rectangles in Golden Ratio proportions seem most pleasing to us, as do triangles with Phi proportions. Buildings constructed using the Golden Ratio tend to be more aesthetically pleasing. The Parthenon makes great use of the Golden Ratio, as does the Great Pyramid at Giza, many Gothic cathedrals, as well as many modern buildings. The Christian Cross is often represented so that the horizontal part of the cross intersects the vertical at the the Golden Ratio proportion. Why do we like things with the Golden Ratio? From The Atlantic: “According to Adrian Bejan, professor of mechanical engineering at Duke University, the human eye is capable of interpreting an image featuring the golden ratio faster than any other….Whether intentional or not, the ratio represents the best proportions to transfer to the brain. This is the best flowing configuration for images from plane to brain and it manifests itself frequently in human-made shapes that give the impression they were ‘designed’ according to the golden ratio,” said Bejan. Here are some examples of the Golden Ratio from nature: “The number of petals in a flower consistently follows the Fibonacci sequence. Famous examples include the lily, which has three petals, buttercups, which have five (pictured below), the chicory’s 21, the daisy’s 34, and so on (all Fibonacci numbers).” (from Gizmodo). Seed heads in daisies and sunflowers follow the Fibonacci sequence (e.g., 21 rows in one direction, 34 in the other direction): Branching in trees typically follow Fibonacci sequence: Animal bodies: “The measurement of the human navel to the floor and the top of the head to the navel is the Golden ratio. But we are not the only examples of the Golden ratio in the animal kingdom; dolphins, starfish, sand dollars, sea urchins, ants and honeybees also exhibit the proportion.” (from Live Science). The size of the hand vs. forearm vs. whole arm also fits the ratio. A DNA molecule measures 34 angstroms by 21 angstroms at each full cycle of the double helix spiral. In the Fibonacci series, 34 and 21 are successive numbers. The arrangement of a pine cone: “The spiral pattern of the seed pods spiral upward in opposite directions. The number of steps the spirals take tend to match Fibonacci numbers.” (From Live Science) Shells often reflect Fibonacci sequences: The shape of Hurricanes: Among other things. Have a nice Monday!

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project Download Circle the letter that corresponds to the correct answer Document related concepts Meteorology Chapter 6 Worksheet 2 Name: ________________________________ Circle the letter that corresponds to the correct answer 1) A steep pressure gradient: a. produces light winds. b. produces strong winds. c. is only possible in the tropics. d. would be depicted by widely spaced isobars. 2) What do isobars represent on a map? a. lines connecting points of equal air pressure b. the lowest pressures on the map c. the highest pressures on the map d. areas of convergence in the upper atmosphere 3) The Coriolis effect occurs because of this characteristic of the earth: a. its magnetic field. b. its atmosphere. c. its rotation. d. its dense core. 4) Horizontal variations in air pressure cause a force which makes the wind blow. These pressure variations are caused by: a. warm temperatures in the stratosphere. b. greenhouse effect. c. non‐circular shape of Earth. d. Earth's rotation. e. uneven heating of the earth's surface. 5) Circulations in the earth's atmosphere are fundamentally caused by: a. heating of the ozone layer. b. frontal storm systems. c. ocean currents. d. gravity. e. temperature contrasts between different locations. 6) The overall strength of a circulation system is determined by: a. the latitude. b. no one factor is more important than the others. c. friction between the ground and the air. d. its pressure gradient. e. air temperature. 7) A plane takes off from City A headed for City B, located directly to the north. The pilot flies directly north, but arrives at a city some distance to the west of City B. What can be said of the airplane? a. It probably has a broken compass. b. It was blown off course by upper atmospheric winds. c. It was flying in the Northern Hemisphere. d. It was flying in the Southern Hemisphere. 8) Refer to the map above. The black lines on the map are called ________ and they represent lines of equal ________. a. isobars; pressure b. isotherms; temperature c. isotherms; pressure d. isodrosotherms; humidity 9) Refer to the map above. Which of the following areas has the highest pressure gradient? a. Southwestern Texas b. Southern California c. Southern Florida d. Lake Michigan/Southeastern Wisconsin 10) Refer to the map above. Which of the following areas is most likely to be experiencing rain or other significant weather? a. the Great Lakes region b. the Southwest c. Western Canada d. the Pacific Northeast 11) Refer to the map above. What best explains the high wind speeds found immediately around the low pressure center (L)? a. the dramatically lower temperatures in the area b. increased friction c. the comparatively high pressure gradient in the area d. the higher humidity associated with low pressures 12) Which of these factors influence the magnitude of the Coriolis force? a. wind direction b. latitude c. wind speed d. both wind speed and latitude 13) As seen by an observer on Earth, the Coriolis effect is an illusion; no deflection can actually be measured. a. false b. false, but only near the poles c. true, but only near the poles d. true 14) The Coriolis effect is important only for motions that: a. do not involve a pressure gradient. b. cover short distances. c. are slow. d. are near the earth's surface. e. cover long distances. 15) The Coriolis effect influences the wind by: a. decreasing the wind speed. b. changing the direction of the wind. c. increasing the wind speed. d. starting the air motion. 16) With respect to the Coriolis force, which association is NOT correct? a. Northern Hemisphere — deflection to the right of the wind's original direction b. North Pole — strongest deflection c. Low wind speeds — strongest deflection d. deflection — always at a 90 degree angle to the direction of air flow 17) Upper air winds: a. are greatly influenced by friction. b. are generally faster than surface winds. c. are unaffected by the Coriolis force. d. do not influence surface weather. 18) The wind speed normally increases with height in the layer of air next to the ground. This illustrates the fact that: a. friction is present only close to the ground. b. the lowest part of the atmosphere is turbulent. c. temperature decreases with height. d. pressure decreases with height. 19) The geostrophic wind concept is most like the real atmospheric winds: a. in an anticyclone. b. near the surface. c. near the equator. d. in a cyclone. e. at high altitudes. 20) When geostrophic conditions exist in the atmosphere, the net force on the moving air is: a. called a centrifugal force. b. zero. c. large when the wind speed is slow. d. called a centripetal force. e. large since the wind speed is fast. 21) The geostrophic wind describes a situation where the air moves: a. very fast. b. upward. c. from pole to equator. d. very slowly. e. parallel to the isobars. 22) What does Buys Ballot's Law state? a. If you stand with your back to the wind, there is low pressure on your left and high pressure on the right. b. If you stand with your back to the wind, there is low pressure on your right and high pressure on your right. c. If you stand with your back to the wind, there is low pressure directly in front of you. d. If you stand facing into the wind and you are facing north, the wind is geostrophic. 23) A cyclone is generally defined by meteorologists as: a. an area of high pressure. b. an area of low pressure. c. an intense, violent storm. d. a tornado on the ground. Circle “T” if the statement is true or “F” if the statement is false T F 24) The most important force causing the air's motion is due to the earth's rotation. T F 25) The speed of the wind at a place is primarily determined by the barometric pressure at that place. T F 26) A steep pressure gradient indicates strong winds. T F 27) The sea breeze is a simple thermal circulation that does not involve a pressure gradient. T F 28) The most fundamental reason for all atmospheric motions is the non‐uniform heating of the earth by the Sun. T F 29) The main cause of the sea breeze is the unequal heating of land and water. T F 30) Vertical air movement is necessary for the creation of a sea breeze. T F 31) The Coriolis effect causes all moving objects to deflect to their right in the northern hemisphere. T F T F T F T F 32) The Coriolis effect only applies to atmospheric motions; aircraft, rockets, people, etc. are not influenced. 33) An isobar is a line connecting points of equal humidity. 34) The Coriolis effect is strongest at the equator and diminishes in strength poleward. 35) Gradient winds follow a curved path. Answer the following questions 36) The tendency of a particle to travel in a straight line creates an imaginary outward force called ________ acceleration. 37) Name the three forces that act to cause the air's motion. 38) What is the fundamental cause of horizontal pressure differences in the atmosphere?

• Quadratic Expressions, Rectangles and Squares • Absolute Value, Square Roots and Quadratic Equations • The Graph Translation Theorem • Graphing 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 • Completing the Square • Fitting a Quadratic Model to Data • The Quadratic Formula • Analyzing Solutions to Quadratic Equations • Solving Quadratic Equations and Inequalities Quadratic – quadratus (Latin) , ‘to make + 𝑏𝑥 + 𝑐 − 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 + 𝑏𝑥 + 𝑐 = 0 − 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 f 𝑥 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 − 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 Standard form of a quadratic: 𝑎𝑥2 + 𝑏𝑥 + 𝑐 Quadratic expressions from Rectangles and Suppose a rectangular swimming pool 50 m by 20 m is to be built with a walkway around it. If the walkway is w meters wide, write the total area of the pool and walkway in standard form. Write the area of the square with sides of length 𝑥 + y in Binomial Square Theorem For all real numbers x and y, 𝑥 + 𝑦 2 + 2xy + 𝑦2 𝑥 − 𝑦 2 = 𝑥2 − 2xy + 𝑦2 Note: When discussing this, ask students whether any real-number values of the variable give a negative value to the expression. [ The square of any real number is nonnegative]. Have students give quadratic expressions for the areas 1. The largest possible circle inside a square whose side 2. The largest possible square inside a circle whose radius 1. Evaluate each of the following. 42, −4 2, 9.32, −9.3 2 2. Find a value of x that is a solution to 𝑥2 = 𝑥. 3. Find a value of x that is not a solution to 𝑥2 = 𝑥. Absolute Value – Square Root Theorem For all real numbers x, 𝑥2 = 𝑥 Solve 𝑥2 = 40 A square and a circle have the same area. The square has side 10. What is the radius of the circle? The Existence of Irrational Numbers Prove that 2 cannot be written as a simple fraction. Graphs and Translations Consider the graphs of 𝑦1 = 𝑥2 and 𝑦2 = 𝑥 − 8 2 What transformation maps the graph of the first function onto the graph of the second? Graph – Translation Theorem In a relation described by a sentence in x and y, the following two processes yield the same graph: 1. replacing 𝑥 by 𝑥 − ℎ and 𝑦 by 𝑦 − 𝑘 2. applying the translation 𝑇ℎ,𝑘 to the graph of the original relation. Find an equation for the image of the graph of 𝑦 = 𝑥 under the The image of the parabola 𝑦 = 𝑎𝑥2 under the translation 𝑇ℎ,𝑘 is the parabola with the equation 𝑦 − 𝑘 = 𝑎 𝑥 − ℎ 2 a. Sketch the graph of 𝑦 − 7 = 3 𝑥 − 6 2 b. Give the coordinates of the vertex of the parabola c. Tell whether the parabola opens up or down d. Give the equation for the axis of symmetry. Graphing 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 Suppose ℎ = −16𝑡2 + 44𝑡 + 5 a. Find ℎ when 𝑡 = 0, 1, 2 𝑎𝑛𝑑 3 b. Explain what each pair 𝑡, ℎ tells you about the height of the ball. c. Graph the pairs 𝑡, ℎ over the domain of the function. Note: Two natural questions about the thrown ball are related to questions about this parabola. 1. How high does the ball get? The largest possible value of h. 2. When does the ball hit the ground? ℎ = − 𝑔𝑡2 + 𝑣0 𝑡 + ℎ0 • 𝑔 is a constant measuring the acceleration due to gravity • 𝑣0 is the initial upward velocity • ℎ0 is the initial height • the equation represents the height ℎ of the ball off the ground at time • Going back to ℎ = −16𝑡2 + 60𝑡 + 5, find the maximum height of the ball. • Rewrite the equation 𝑦 = 𝑥2 + 10𝑥 + 8 in vertex form. Locate the vertex of the parabola. • Suppose 𝑓 𝑥 = 3𝑥2 + 12𝑥 + 16 a. What is the domain of 𝑓? b. What is the vertex of the graph? c. What is the range of 𝑓? Fitting a Model to Data The number of handshakes ℎ needed for everyone in a group of 𝑛 people, 𝑛 ≥ 2, to shake the hands of every other person is a quadratic function of 𝑛. Find three points of the function relating ℎ and 𝑛. Use these points to find a formula for this function. The Angry Blue Bird Problem What if Blue Bird’s flight path is described by the function ℎ 𝑥 = −0.005𝑥2 + 2𝑥 + 3.5 Where is Blue Bird when she’s 8 feet high? The Quadratic Formula If 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 𝑎𝑛𝑑 𝑎 ≠ 0 , 𝑡ℎ𝑒𝑛 How was it derived? Solve 3𝑥2 + 11𝑥 − 4 = 0 The 3-4-5 right triangle has sides which are consecutive integers. Are there any other right triangles with this Challenge: Find a number such that 1 less than the number divided by the reciprocal of the number is equal How Many Real Solutions Does a Quadratic Equation Suppose 𝑎, 𝑏, 𝑎𝑛𝑑 𝑐 are real numbers with 𝑎 ≠ 0. Then the equation 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 has i. two real solutions if 𝑏2 − 4𝑎𝑐 > 0. ii. one real solution if 𝑏2 − 4𝑎𝑐 = 0. iii. two complex conjugate solutions if 𝑏2 − 4𝑎𝑐 < 0. Determine the nature of the roots of the following equations. Then solve. − 12𝑥 + 9 = 0 b. 2𝑥2 + 3𝑥 + 4 = 0 − 3𝑥 − 9 = 0 • Transform the quadratic equation into standard form if necessary. • Factor the quadratic expression. • Apply the zero product property by setting each factor of the quadratic expression equal to 0. Zero Product Property – If the product of two real numbers is zero, then either of the two is equal to zero or both numbers are equal to zero. • Solve each resulting equation. • Check the values of the variable obtained by substituting each in the original HOW TO SOLVE? 1. find the "=0" points 2. in between the "=0" points, are intervals that are either greater than zero (>0), or less than zero (<0) 3. then pick a test value to find out which it is (>0 or <0) Here is the plot of 𝑦 = 𝑥2 − 𝑥 − 6 The equation equals zero at -2 and 3 The inequality "<0" is true between -2 and 3. 1. Find the solution set of 𝑥2 − 5𝑥 − 14 > 0. 2. Graph 𝑦 > 𝑥2 − 5𝑥 − 14 3. A stuntman will jump off a 20 m building. A high-speed camera is ready to film him between 15 m and 10 m above the ground. When should the camera film him?

So there’s no better way to understand the “laws” of thermodynamics than to actually get a textbook and work some problems! But I’m not going to hold it against you if that doesn’t seem like the best use of your time. Instead then let me attempt to summarize the first and second laws as best I can with an eye to the philosophically salient points required to appreciate their application to life. This post will concern itself with the first law and energy while the following post will discuss the second law and entropy. The first law is conservation of energy. It says that the amount of energy in an isolated system cannot change. You may have heard “energy can be neither created nor destroyed.” This is another way of saying that the total quantity of energy in a system cannot change if energy cannot enter or leave from the outside. "System" means any arbitrary portion of the world you care to define, but you have to pick boundaries and you have to know whether radiation or mass can cross those boundaries if you want to be able to calculate things, so you had best to choose your system boundaries carefully! Once this is done though, thermodynamics provides you with a way of predicting what will happen to your system provided you know enough about it. A mathematically convenient way of stating that a quantity is constant is stating that the total change of this quantity is zero, or that ∆E=0. Once you can write that down then you can make an equation that connects conditions in the past to conditions in the future, because you know that the energy will be the same at all times. You’d be amazed at how little we’d be able to predict about nature if this statement wasn't true. The system, its boundary, and the surroundings are the only objects that exist in the thermodynamic formalism. Ok so that’s what the first law says but what is energy, anyway? We hear a lot about different kinds of “energies” but the simplest definition of energy comes from the work-energy theorem: Energy is what is capable of doing work. What is work? Work is moving mass over distance, like hauling timber or lifting a coffee cup. If we put all that together then we arrive at the statement: “The total capacity to move mass through space in an isolated system is constant.” I think this statement is an adequate description of the first law. But let me clarify a few points. First, the definition of energy as “how much mass can be moved how far” underscores the fact that energy is a conceptual abstraction. The four forces (but I prefer the word interactions) of nature are understood as the fundamental physical “givens” responsible for this total energy. This is just saying that gravity or electromagnetism, which are the interactions relevant to biology, are the things that “cause” the movement in the mechanistic sense. What we refer to as energy is a way of quantifying the potential for motion that exists because of these interactions. What makes “energy” such a grand unifying abstraction is precisely it’s mathematical property of being a constant quantity. Feynman quipped the following about the first law (my emphasis added): ...It states that there is a certain quantity, which we call energy that does not change in manifold changes which nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity, which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number, and when we finish watching nature go through her tricks and calculate the number again, it is the same. Two points about this quote: Energy is a numerical quantity. I would restate that point by saying that energy is a mathematical abstraction. It is not “out there in the world” in the way that trees and electromagnetism are “out there.” I’m fond of the assertion that energy is what is, not because energy is a fundamental substratum or essence of “reality” in any sense but rather precisely for the opposite reason that it is a mathematical abstraction which is invariant in time. So that’s the first point--Energy is not a physical thing but a mathematical idea which was “thingified” by language because it was found to have the immensely useful property of never changing no matter how the actual matter under consideration was found to move. The first point is related to the second, which is that the fact of conservation of energy is not a “mechanism.” It is a rule which “mechanics” obeys but it does not tell you how anything happens. So now a brief digression to explain this point. The discipline of mechanics, whether classical or quantum, is very much in the business of telling us how things happen. When I apply a force to a ball by kicking it I “cause” the ball to accelerate in a mechanistic way, and if someone asks me “how” I made the ball accelerate I can honestly respond “by kicking it.” This is how we would normally render Newton’s second law, f=ma, into language: A force causes a mass to accelerate...so I applied a force with my foot and off the thing went. (Note that it's rather odd to render an equal sign as "cause," but this is how it is generally interpreted at any rate.) Of course, this is not an adequate response to the question of why I kicked the ball and we don’t expect mechanics to answer this question. With this rough distinction between mechanistic how’s and purposive why’s in mind, we come to a platitude repeated in high school courses that I truly despise: Science tells us how things happen, but it requires religion, philosophy, or some form of ethics to discuss why things happen. I have no idea what does or does not require philosophy, but the assertion about science is only true if we restrict “science” to the disciplines of classical and quantum mechanics. But this is not even all of physics, much less all of science! As I hope you will come to understand, the laws of thermodynamics are about why things happen naturally or, to use the jargon, spontaneously. For the time being suffice it to say that if they explain anything, they explain why a process occurs because they certainly don't explain how anything happens. This is why thermodynamics and mechanics evolved as distinct disciplines until their unification through statistical mechanics (more about this in later posts) and also why mechanics does not deal explicitly with time-directed, irreversible processes. So “science” has never been in the business of avoiding questions about why things happen and it should not avoid them because its real job is to provide adequate explanations of natural phenomena. Nothing that leaves one with no understanding of why a process occurs could be an adequate explanation. However, biology and the theory of evolution in particular, as I wrote about before, have been particularly damaged by this view that “science” must restrict itself to providing only mechanistic “hows.” In fact, while I’m digressing, in case you haven’t figured it out by now, I don’t believe in Science as such at all. There are only the myriad things people do and measure under more or less controlled conditions in more or less clever or technical ways with concepts that are more or less quantifiable. People who invoke the spirit of Science with a capital S (White lab coats on white males and “pure facts” rendered in the antiseptic UV light of enlightenment reason) most often merely wish to claim blanket authority for ideas which, in the spirit of science, ought to stand on their own. At any rate that’s what I think you ought to know about the first law. Energy became the central concept of thermodynamics because it has the enormously useful property of being constant in time. How useful? So useful they call it the first law of thermodynamics, and the predictability of the final state of any natural process is grounded in our capacity to write down equations which connect the past and the future. These equations are all in some way derived from the constancy in time of energy. But it is not considered to be the “cause” of anything--gravity and voltages and nuclear forces and so on provide the "how" cause of things. Furthermore, this “law,” or as Feynman says “strange fact,” does not by itself explain why a process does or does not occur. That’s really the second law’s territory, which we’ll get to next time. But understanding entropy, which the second law tells us is the quantity that can only increase, requires this preliminary understanding of energy, the quantity that cannot change. Finally, once we understand the basic ideas of thermo, then we can begin to see how the interaction between that which can’t change and that which can only increase creates these beautiful natural patterns from which we grew. Fractal pattern indicating self-organization via dissipative flows Thumbnail image: A forest fire represents a massive transfer of energy from system to surroundings, but the total energy of the universe is unchanged during this process.

Intro to Series A mathematical series is a gotten listing of things, frequently numbers. Often the numbers in a series are specified in regards to a previous number in the checklist. Secret TakeawaysSecret PointsThe variety of purchased aspects (potentially unlimited) is called the size of the series. Unlike a collection, order issues, as well as a specific term can show up numerous times at various settings in the sequence.A math series is one in which a term is gotten by including a consistent to a previous regard to a series. So the n th term can be defined by the formula a_n = a _ n-1 + d A geometric series is one in which a regard to a series is gotten by increasing the previous term by a continuous. It can be explained by the formula a_n=r \ cdot a _ Secret Terms In maths, a series is a gotten checklist of things. Like a collection, it has participants (additionally called terms or aspects). The variety of gotten components (perhaps unlimited) is called the size of the series. Unlike a collection, order issues, as well as a specific term can show up several times at various placements in the series. As an example, (M, A, R, Y) is a series of letters that varies from (A, R, M, Y) , as the getting issues, as well as (1, 1, 2, 3, 5, 8) , which includes the number 1 at 2 various placements, is a legitimate series. Series can be limited, as in this instance, or unlimited, such as the series of all also favorable integers (2, 4, 6, \ cdots) Limited series are often referred to as words or strings and also limitless series as streams. Instances as well as Symbols Limitless as well as limited Series An even more official interpretation of a limited series with terms in an established S is a feature from \ left \ 1, 2, \ cdots, n \ ideal \ to S for some n> 0 An unlimited series in S is a feature from \ left \ to S For instance, the series of prime numbers (2,3,5,7,11, \ cdots) is the feature 1 \ rightarrow 2, 2 \ rightarrow 3, 3 \ rightarrow 5, 4 \ rightarrow 7, 5 \ rightarrow 11, \ cdots A series of a limited size n is additionally called an n -tuple. Limited series consist of the vacant series (\ quad) that has no aspects. Much of the series you will certainly come across in a math training course are generated by a formula, where some procedure(s) is executed on the previous participant of the series a _ n-1 to provide the following participant of the series a_n These are called recursive series. A math (or direct) series is a series of numbers in which each brand-new term is determined by including a continuous worth to the previous term. An instance is (10,13,16,19,22,25) In this instance, the initial term (which we will certainly call a_1 is 10 , and also the typical distinction ( d -- that is, the distinction in between any type of 2 nearby numbers-- is 3 The recursive interpretation is for that reason \ displaystyle a_n=a _ +3, a_1=10 An additional instance is (25,22,19,16,13,10) In this instance a_1 = 25 , as well as d=-3 The recursive interpretation is for that reason \ displaystyle In both of these instances, n (the variety of terms) is 6 A geometric series is a checklist in which each number is produced by increasing a consistent by the previous number. An instance is (2,6,18,54,162) In this instance, a_1=2 , as well as the usual proportion ( r -- that is, the proportion in between any kind of 2 surrounding numbers-- is 3. As a result the recursive interpretation is a_n=3a _ , a_1=2 One more instance is (162,54,18,6,2) In this instance a_1=162 , and also \ displaystyle r=\ frac 1 3 As a result the recursive formula is \ displaystyle In both instances n=5 A specific interpretation of a math series is one in which the n th term is specified without referring to the previous term. This is better, since it implies you can locate (as an example) the 20th term without discovering every one of the various other terms in between. To locate the specific meaning of a math series, you start drawing up the terms. Presume our series is t_1, t_2, \ dots The initial term is constantly t_1 The 2nd term increases by d , therefore it is t_1+d The 3rd term increases by d once again, therefore it is (t_1+d)+d, or simply put, t_1 +2 d So we see that: \ displaystyle \ start line up t_1 &= t_1 \ \ t_2 &= t_1+d \ \ t_3 &= t_1 +2 d \ \ t_4 &= t_1 +3 d \ \ & \ vdots \ end and more. From this you can see the generalization that: t_n = t_1+(n-1)d which is the specific meaning we were trying to find. The specific interpretation of a geometric series is acquired in a comparable method. The initial term is t_1 ; the 2nd term is r times that, or t_1r ; the 3rd term is r times that, or t_1r ^ 2 ; and so forth. So the basic regulation is: t_n=t_1 \ cdot r ^ The General Regard To a Series Offered terms in a series, it is commonly feasible to locate a formula for the basic regard to the series, if the formula is a polynomial. Secret TakeawaysTrick PointsGiven terms in a series created by a polynomial, there is a technique to identify the formula for the polynomial.By hand, one can take the distinctions in between each term, then the distinctions in between the distinctions in terms, and so on. If the distinctions ultimately end up being continuous, then the series is created by a polynomial formula.Once a consistent distinction is accomplished, one can resolve formulas to produce the formula for the polynomial.Key Terms Offered numerous terms in a series, it is in some cases feasible to discover a formula for the basic regard to the series. Such a formula will certainly generate the n th term when a worth for the integer n is taken into the formula. This reality can be identified by discovering whether the computed distinctions ultimately come to be continuous if a series is created by a polynomial. Think about the series: 5, 7, 9, 11, 13, \ dots The distinction in between 7 and also 5 is 2 The distinction in between 7 and also 9 is additionally 2 Actually, the distinction in between each set of terms is 2 Considering that this distinction is continuous, and also this is the initial collection of distinctions, the series is provided by a first-degree (direct) polynomial. Expect the formula for the series is provided by an+b for some constants a as well as b Then the series appears like: a+b, 2a+b, 3a+b, \ dots The distinction in between each term as well as the term after it is a In our instance, a=2 It is feasible to fix for b making use of among the terms in the series. Utilizing the initial number in the series and also the initial term: \ displaystyle So, the n th regard to the series is provided by 2n +3 Expect the n th regard to a series was provided by an ^ 2+bn+c Then the series would certainly resemble: a+b+c, 4a +2 b+c, 9a +3 b+c, \ dots This series was developed by connecting in 1 for n , 2 for n , 3 for n , and so on. If we begin at the 2nd term, as well as deduct the previous term from each term in the series, we can obtain a brand-new series comprised of the distinctions in between terms. The very first series of distinctions would certainly be: 3a+b, b+5a, 7a+b, \ dots Currently, we take the distinctions in between terms in the brand-new series. The 2nd series of distinctions is: 2a, 2a, 2a, 2a, \ dots The computed distinctions have actually assembled to a consistent after the 2nd series of distinctions. This indicates that it was a second-order (square) series. Functioning backwards from this, we can locate the basic term for any type of square series. Take into consideration the series: 4, -7, -26, -53, -88, -131, \ dots The distinction in between -7 and also 4 is -11 , as well as the distinction in between -26 as well as -7 is -19 Locating all these distinctions, we obtain a brand-new series: -11, -19, -27, -35, -43, \ dots This checklist is still not continuous. Nonetheless, locating the distinction in between terms again, we obtain: -8, -8, -8, -8, \ dots This truth informs us that there is a polynomial formula explaining our series. Because we needed to do distinctions two times, it is a second-degree (square) polynomial. We can locate the formula by understanding that the consistent term is -8 , which it can additionally be revealed by 2a As a result a=-4 Next we keep in mind that the very first thing in our very first listing of distinctions is -11 , however that generically it is meant to be 3a+b , so we need to have 3( -4 )+b=-11 , and also b=1 Ultimately, note that the initial term in the series is 4 , as well as can likewise be revealed by a+b+c = -4 +1+c So, c=7 , as well as the formula that creates the series is -4 a ^ 2+b +7 c General Polynomial Series This technique of discovering distinctions can be encompassed locate the basic regard to a polynomial series of any kind of order. For greater orders, it will certainly take a lot more rounds of taking distinctions for the distinctions to end up being consistent, and also extra back-substitution will certainly be needed in order to address for the basic term. General Regards To Non-Polynomial Series Some series are produced by a basic term which is not a polynomial. For instance, the geometric series 2, 4, 8, 16, \ dots is offered by the basic term 2 ^ n Taking distinctions will certainly never ever result in a continuous distinction since this term is not a polynomial. General regards to non-polynomial series can be located by monitoring, as above, or by various other ways which are past our range in the meantime. Offered any kind of basic term, the series can be produced by connecting in succeeding worths of n Collection and also Sigma Symbols Sigma symbols, represented by the uppercase Greek letter sigma \ left (\ Sigma \ right ), is utilized to stand for summations-- a collection of numbers to be totaled. Secret TakeawaysSecret PointsA collection is a summation executed on a checklist of numbers. Each term is contributed to the following, causing an amount of all terms.Sigma symbols is utilized to stand for the summation of a collection. In this kind, the resources Greek letter sigma \ left (\ Sigma \ right) is made use of. The series of terms in the summation is stood for in numbers listed below as well as over the \ Sigma sign, called indices. The most affordable index is composed listed below the sign as well as the biggest index is created above.Key Terms Summation is the procedure of including a series of numbers, leading to an amount or overall. Any type of intermediate outcome is a partial amount if numbers are included sequentially from left to right. The numbers to be summed (called addends, or occasionally summands) might be integers, reasonable numbers, genuine numbers, or complicated numbers. For limited series of such components, summation constantly creates a distinct amount. A collection is a checklist of numbers-- like a series-- yet rather than simply detailing them, the plus indications show that they need to be built up. For instance, 4 +9 +3 +2 +17 is a collection. This certain collection amounts to 35 One more collection is 2 +4 +8 +16 +32 +64 This collection amounts to 126 One means to compactly stand for a collection is with sigma symbols , or summation symbols , which appears like this: \ displaystyle \ amount _ n=3 ^ The primary icon seen is the uppercase Greek letter sigma. It shows a collection. To "unbox" this symbols, n=3 stands for the number at which to begin counting ( 3 , as well as the 7 stands for the factor at which to quit. For each and every term, plug that worth of n right into the provided formula ( n ^ 2 . This specific formula, which we can review as "the amount as n goes from 3 to 7 of n ^ 2 ," indicates: \ displaystyle Much more typically, sigma symbols can be specified as: \ displaystyle In this formula, i stands for the index of summation, x_i is an indexed variable standing for each succeeding term in the collection, m is the reduced bound of summation, and also n is the top bound of summation. The" i = m under the summation icon indicates that the index i starts equivalent to m The index, i , is incremented by 1 for each and every succeeding term, quiting when i=n One more instance is: \ displaystyle This collection amounts to 90. So we can create: \ displaystyle \ amount _ ^ 6 (i ^ 2 +1)=90 Various Other Types of Sigma Symbols When these are clear from context, casual writing occasionally leaves out the interpretation of the index and also bounds of summation. As an example: \ displaystyle \ amount x_i ^ 2=\ amount _ ^ n x_i ^ 2 A recursive interpretation of a feature specifies its worths for some inputs in regards to the worths of the exact same feature for various other inputs. Trick TakeawaysTrick PointsIn mathematical reasoning as well as computer technology, a recursive meaning, or inductive interpretation, is made use of to specify an item in regards to itself.The recursive meaning for a math series is: a_n=a _ n-1 +d The recursive interpretation for a geometric series is: a_n=r \ cdot a _ n-1 In mathematical reasoning as well as computer technology, a recursive meaning, or inductive interpretation, is made use of to specify a things in regards to itself. A recursive meaning of a feature specifies worths of the feature for some inputs in regards to the worths of the exact same feature for various other inputs. As an example, the factorial feature n! is specified by the policies: 0!=1 (n +1)!=(n +1)n! This interpretation stands due to the fact that, for all n , the recursion ultimately gets to the base instance of 0 For instance, we can calculate 5! by understanding that 5!=5 \ cdot 4! , which 4!=4 \ cdot 3! , which 3!=3 \ cdot 2! , which 2!=2 \ cdot 1!, which: \ displaystyle \ start line up 1! &=1 \ cdot 0! \ \ &= 1 \ cdot 1 \ \ &=1 \ end Placing this completely we obtain: \ displaystyle Recursive Solutions for Series When going over math series, you might have observed that the distinction in between 2 successive terms in the series can be created in a basic method: a_n=a _ +d The above formula is an instance of a recursive formula given that the n th term can just be computed by thinking about the previous term in the series. Contrast this with the formula: a_n=a_1+d(n-1). In this formula, one can straight determine the nth-term of the math series without recognizing the previous terms. Relying on just how the series is being made use of, either the non-recursive one or the recursive interpretation may be better. A recursive geometric series complies with the formula: a_n=r \ cdot a _ A used instance of a geometric series includes the spread of the influenza infection. Mean each contaminated individual will certainly contaminate 2 even more individuals, such that the terms adhere to a geometric series. Utilizing this formula, the recursive formula for this geometric series is: a_n=2 \ cdot a _ n-1 Recursive formulas are incredibly effective. One can exercise every term in the collection simply by recognizing previous terms. As can be seen from the instances over, exercising as well as making use of the previous term a _ n − 1 can be a much easier calculation than exercising a _ n from the ground up utilizing a basic formula. When making use of a computer system to control a series could indicate that the estimation will certainly be ended up promptly, this implies that making use of a recursive formula.

Chaos synchronization of the Sprott N System with parameter. Since synchronization of chaotic systems was first introduced by Fujisaka and Yamada and Pecora and Carroll . Due to the importance and applications of coupled systems, ranging from chemical oscillators, coupled neurons, coupled circuits to mechanical oscillators, various synchronization schemes have been proposed by many scientists from different research fields [3-10]. Recently, a new type of chaotic synchronization-full state hybrid projective synchronization(FSHPS) in continuous-time chaotic and hyper-chaotic systems based on the Lyapunov's direct method is presented and investigated by wen,many notable results and a series of important applications to security communication regarding FSHPS has been presented in Refs [12-14]. We organize this paper as follows. In Section 2, the chaotic characteristic of the Sprott N autonomous system with parameter is studied by theoretical analysis and numerical simulation. In section 3, the scheme of full state hybrid projective synchronization(FSHPS) is given.A proper Smarandache controller is designed and the synchronization of the system is achieved under it. [FIGURE 1 OMITTED] [section]2. Dynamical behavior A series of three dimensional autonomous systems is presented by J.C. Sprott in 1994 , in this paper, the N system of those is taken as example. The governing equations of the Sprott N system are: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1) The equilibrium of the system (1) is P(-0.25,0.0.5).The Lyapunov exponents of the system are LEs=(0.076,0,-2.076),which shows the system is chaotic. In order to get abundance dynamical behavior of the system, the parameter /3 is leaded to the system. The governing equations of the Sprott N system with parameter [beta] can be described as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2) where x = ([x.sub.1], [x.sub.2], [x.sub.3]) is the state variable. The initial conditions are ([x.sub.1](0), [x.sub.2](0), [x.sub.3](0))=(1,5,2), when the parameter [beta] = 1.9, the system exits a chaotic attractor. The chaotic attractor in three-dimensional phase space is illustrated in Fig 1. For this system, bifurcation can easily be detected by examining graphs of abs(z)versus the control parameter [beta]. The dynamical behavior of the system (2) can be characterized with its Lyapunov exponents which are computed numerically. The bifurcation diagram and the Lyapunov exponents spectrum are showed in Fig 2. [FIGURE 2 OMITTED] [section]3. Chaos synchronization base on FSHPS We recall a class of automomous chaotic flows in the form of x(t) = F(x), (3) x = [([x.sub.1], [x.sub.2],...,[x.sub.n]).sup.T] is the state vector, and F(x) = [([F.sub.1](x), [F.sub.2](x),...,[F.sub.n](x)).sup.T] is continuous nonlinear vector function. We take (3) as the drive system and the response system is given by y(t) = F(y) + u, (4) y = [([y.sub.1],[y.sub.2],...,[y.sub.n]).sup.T] is the state vector, and F(y) = [([F.sub.1](y),[F.sub.2](y),...,[F.sub.n](y)).sup.T] is continuous nonlinear vector function. u = u(x,y) = [([u.sub.1](x,y),[u.sub.2](x,y),...,[u.sub.n](x,y)).sup.T] is the controller to be determined for the purpose of full state hybrid projective synchronization.Let the vector error state be e(t) = y(t) -[alpha]x(t).Thus, the error dynamical system between the drive system (3) and the response system (4) is e(t) = y(t) - [alpha]x(t) = F(x,y) + u, (5) where F(x,y) = F(y) - [alpha]F(x) = [([F.sub.1](y) - [[alpha].sub.1][F.sub.1]](x),[F.sub.2](y) - [[alpha].sub.2][F.sub.2](x),...,[F.sub.n](y) - [[alpha].sub.n][F.sub.n](x)).sup.T]. In the following, we will give a simple principle to select suitable feedback controller a such that the two chaotic or hyper-chaotic systems are FSHPS. If the Lyapunov function candidate V is take as: V = 1/2[e.sup.T][P.sub.e], (6) where P is a positive definite constant matrix, obviously, V is positive define. One way choose as the corresponding identity matrix in most case. The time derivative of V along the trajectory of the error dynnmical system is as follows V = [e.sup.T]P(u + F), (7) Suppose that we can select an appropriate controller u such that V is negative definite. Then, based on the Lyapunov's direct method, the FSHPS of chaotic or hyper-chaotic flows is synchronization under nonlinear controller u . In order to observe the FSHPS of system (2), we define the response system of (2) as follows [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8) where u = [([u.sub.1], [u.sub.2], [u.sub.3]).sup.T] is the nonlinear controller to be designed for FSHPS of two Sprott N chaotic systems with two significantly different initial conditions. Define the FSHPS error signal as e(t) = g(t) - cxx(t),i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9) where [alpha] = diag([[alpha].sub.1], [[alpha].sub.2], [[alpha].sub.3]),and [[alpha].sub.1], [[alpha].sub.2] and [[alpha].sub.3] are different desired in advance scaling factors for FSHPS. The error dynnmical system can be written as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10) The goal of control is to find a controller u = [([u.sub.1], [u.sub.2], [u.sub.3]).sup.T] for system (10) such that system (2) and (8) are in FSHPS. We now choose the control functions [u.sub.1], [u.sub.2] and [u.sub.3] as follows [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11) If the Lyapunov function candidate is taken as: V = 1/2([e.sup.2.sub.1] + [e.sup.2.sub.2] + [e.sup.2.sub.3]), (12) The time derivative of V along the trajectory of the error dynnmical system (10) is as follows V = - ([e.sup.2.sub.1] + [e.sup.2.sub.2] + [e.sup.2.sub.3]), (13) Since V is a positive definite function and V is a negative definite function, according to the Lyapunov's direct method, the error variables become zero as time tends to infinity, i.e., [lim.sub.t[right arrow][infinity]] || [y.sub.i] - [[alpha].sub.i][x.sub.i] || = 0, i = 1, 2, 3. This means that the two Sprott A systems are in FSHPS under the controller (11). [FIGURE 3 OMITTED] For the numerical simulations, fourth-order Runge-Kutta method is used to solve the systems of differential equations (2) and (8). The initial states of the drive system and response system are ([x.sub.1](0), [x.sub.2](0), [x.sub.3](0) = (1, 5, 2) and ([y.sub.1](0), [y.sub.2](0), [y.sub.3](0) = (11,15,12) The state errors between two Sprott systems are shown in Fig.3. Obviously, the synchronization errors converge asymptotically to zero and two systems are indeed achieved chaos synchronization. (section)4. Conclusion and discussion In the paper, the problem synchronization of the Sprott N system with parameter is investigated. An effective full state hybrid projective synchronization (FSHPS) controller and analytic expression of the controller for the system are designed. Because of the complete synchronization, anti-synchronization, projective synchronization are all included in FSHPS, our results contain and extend most existing works. But there are exist many interesting and difficult problems left our for in-depth study about this new type of synchronization behavior, therefore, further research into FSHPS and its application is still important and insightful, although it is not in the category of generalized synchronization. Fujisaka H, Yamada T. Stability theory of synchronized motion in coupled-oscillator systems. Prog Theory Phys, 69(1983), No.l, 32-71. Peroca LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett, 64(1990), No.8, 821-4. Kocrev L, Parlitz U. Generalized synchronization, predictability and equivalence of unidirectionally coupled system. Phys Rev Lett, 76(1996), No.ll, 1816-9. Vincent UE, Njah AN, Akinlade O, Solarin ART. Phase synchronization in unidirectionally coupled chaotic ratchets. Chaos, 14(2004), No.4, 1018-25. Liao TL. Adaptive synchronization of two Lorenz systems. Chaos, Solitons and Fractals, 9(1998), No.2, 1555-61. Rosenblum MG, Pikovsky AS, Kurths J. From phase to lag synchronization in coupled chaotic oscillators. Phys Rev Lett, 78(1997), No.22, 4193-6. Voss HU. Anticipating chaotic synchronization. Phys Rev E, 61(2000), No.5, 5115-9. Bai EW, Lonngran EE. Synchronization of two Lorenz systems using active control. Chaos, Solitons and Fractals, 8(1997), No.1, 51-8. Vincent UE. Synchronization of Rikitake chaotic attractor using active control. Phys Lett A, 343(2005), No.2, 133-8. Lu J, Wu X, Han X, Lu J Adaptive feedback synchronization of a unified chaotic system, Phys Lett A, 329(2004), No.2, 327-33. Wen GL, Xu D. Nonliear observer control for full-state projective synchronization in chaotic continuous-time system Chao, Solitions and Fractal, 26(2005), No. 2, 71-7. Xu D, Li Z, Bishop R. Manipulating the scaling factor of projective synchronization in three-dimensional chaotic system. Chaos, 11(2001), No.3, 439-42. Wen GL, Xu D. Observer-based control for full-state projective synchronization of a general class of chaotic maps in ant dimension. Phys Lett A, 333(2004), No.2, 420-5. Li ZG. Xu D. Stability criterion for projective synchronization in three-dimensional chaotic systems. Phys Lett A, 282(2001), No.2, 175-9. Sprott JC. Some simple chaotic flows.Phys Rev E, 50(1994), No.2, 647-650. Xiaojun Liu ([dagger]), Wansheng He ([dagger]), Xianfeng Li([double dagger]), Lixin Yang ([dagger]) ([dagger]) Department of Mathematics and Statistics, Tianshui Normal University, Tianshui, Gansu, 741001, P.R.China ([double dagger]) School of Mathematics, Physics and Software Engineering, Lanzhou Jiaotong University Lanzhou, Gansu. 730070, P.R.China (1) This work is supported by the Gansu Provincial Education Department Foundation 0808-04 and Scientific Research Foundations of Tianshui Normal University of China TSB0818. |Printer friendly Cite/link Email Feedback| |Author:||Xiaojun, Liu; Wansheng, He; Xianfeng, Li; Lixin, Yang| |Date:||Jun 1, 2009| |Previous Article:||Green~ relations and the natural partial orders on U-semiabundant semigroups.| |Next Article:||On the equivalence of some iteration schemes with their errors.|

This is a weekend divergence, but a timely and appropriate topic. For one thing, it’s Super Bowl weekend; and I am putting together the “Statistical Inference Part 3” handout for my research methods course on hypotheses and hypothesis testing. You are probably not aware of this but deflate gate (whether the Patriot’s intentionally deflated footballs) has kicked off a statistical analysis frenzy. Here is one by Sharp Football Analysis that has the attention of a blog I follow. The blog I follow is Flowing Data – which also appropriately refers to one from Regressing called “Why Those Statistics about the Patriot’s Fumbling Are Mostly Junk.” Flowing data and Regressing offer pretty sophisticated commentary on the statistical assumptions made by the author from Sharp Football Analysis. I am not going to attempt to offer any additional statistical commentary. The fact is – regardless of how you look at it there is a difference between the Patriot’s fumbling and the other team’s fumbling in NFL games. It is overestimated in the Sharp Football Analysis, but there is a difference. I want to talk about that for this post. There is a difference. As you know statistical analysis that puts forward the probability of a difference occurring is testing the probability of the data conditional on the assumption that no difference exists. This is the null hypothesis. Again, the null hypothesis is that there is no difference; and the test is trying to obtain: Pr(data | null hypothesis is true) – – that is what a “p” value is testing; the conditional probability that the data would occur if the null hypothesis were true. And in the case of the Patriot’s, there is a low probability (how low depends on how you calculate it and I think Regressing has done a nice job with it) that the data would occur if the null were true. With a small p value we can “reject the null” so to speak. Let’s put aside the underlying assumption of a random sample for now – clearly these are not random samples so the entire approach of using statistics based on sampling distributions to estimate sample error should be questioned. When we reject the null hypothesis we are happy to put forward the “alternate” hypothesis – the hypothesis that “MUST BE” true – that “HAS TO BE TRUE” because the alternate hypothesis is the opposite of the null, the null is wrong as we just rejected it, and we live in a world of non contradiction, so if the null is rejected the alternate is accepted – taken as true. Does this work? Yes, but with limits based on what the alternate is actually saying. Let’s accept that the null is rejected (there is not no difference between the Patriots and other teams in the “effect” – fumbling). I bolded my double negative because that is what we do when we reject the null. We say – there is not no difference, therefore there is a difference (an effect, an association). The alternate is that there is a difference is accepted. Fine. Let’s accept that there is a difference. But that is all the alternate can say. All the alternate says is that there is a difference. The next BIG question is – why is there a difference? In a controlled trial this is easier to answer. Things are controlled, even randomized, the only accepted difference between the conditions is a particular suspected causal agent. But this is not a controlled trial, these teams have not been selected or put together by a randomization. Deflate gate is about one possible causal agent – deflated footballs. But the analyses done, the statistical probabilities reported, are not specifically testing that as an alternate hypothesis. The only thing they are showing is that there is a difference. Since this is not a controlled trial we are left to attempt to “abduct” what the best explanation of the difference might be – but that is not tested at all by the statistics being looked at for deflate gate, and only discussed a tiny bit by the analysts. To accept the alternate (they are different) is NOT to accept a particular explanation of why the alternate occurred (the balls were deflated). There are other perfectly reasonable explanations of the alternate – THE PATRIOTS ARE A BETTER TEAM – and the better team will score more points, have less points scored against them, create more turnovers, PRODUCE LESS TURNOVERS. The numbers being produced do not justify one of these explanations for the alternate over the other – that is the point of a discussion. But what I have been reading are completely biased explanations for the alternate. As an example, when discussing great quarterbacks it is accepted that they throw significantly less interceptions, we reject the null, we accept the alternate that they are different. BUT, we use that data to explain why they are great quarterbacks, we do not assume they must be doing something with the footballs to throw less interceptions. So then why not interpret a team with less fumbles as evidence that they are a better team? Here I defend Sharp a bit. Two things in the defense of Sharp Football Analysis. First, there is a reason to suspect deflated footballs as a possible explanation – footballs were found under pressure in the AFC Championship game. So it is not that this possible explanation comes out of the blue – there is a valid reason to put it forward as a possible explanation. But being forward as a possible explanation does not make it THE explanation. Second, they do put forward a series of other explanations (at the end of the post) that they discuss off hand and with sense of disregard: “Could the Patriots be so good that they just defy the numbers? As my friend theorized: Perhaps they’ve invented a revolutionary in-house way to protect the ball, or perhaps they’ve intentionally stocked their skill positions with players who don’t have a propensity to fumble. Or perhaps still, they call plays which intentionally result in a lower percentage of fumbles. Or maybe its just that they play with deflated footballs on offense. It could be any combination of the above.” “But regardless of what, specifically, is causing these numbers, the fact remains: this is an extremely abnormal occurrence and is NOT simply random fluctuation.” We can agree – not simple random fluctuation. But that does not mean we can infer deflated balls. They disregard these other explanations but with no real justification other than the occurrence of deflated balls at one game. We can list a whole host of other reasons that explain the statistical difference. After all, a team will be better than the rest at things – and that team will appear extreme relative to the mean – and no one is saying it is due to random fluctuation. But just because it is not random fluctuation we cannot claim that it is because footballs were deflated. The fact is, there is an extremely complex causal network for the difference and to claim deflated balls as the cause first requires all other causal paths to the effect of interest be considered with data to rule them out prior to inference of one particular cause. I am all for listening to the data. But let’s keep in mind that without our guidance, the data has very little to say.

Board Paper of Class 10 2019 Maths Abroad(Set 3) - Solutions (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided into four sections – A, B, C and D. (iii) Section A comprises 6 questions of 1 mark each. Section B contains 6 questions of 2 marks each. Section C contains 10 questions of 3 marks each. Section D contains 8 questions of 4 marks each. (iv) There is no overall choice. However, an internal choice has been provided in two questions of 1 mark, two questions of 2 marks, four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternative in all such questions. (v) Use of calculators is not permitted. - Question 1 Which term of the A.P. −4, −1, 2, ... is 101? VIEW SOLUTION - Question 2 Express (sin 67° + cos 75°) in terms of trigonometric ratios of the angle between 0° and 45°. VIEW SOLUTION - Question 3 Find the value of k for which the quadratic equation kx (x − 2) + 6 = 0 has two equal roots. VIEW SOLUTION - Question 4 Find a rational number between and . Write the number of zeroes in the end of a number whose prime factorization is 22 × 53 × 32 × 17. VIEW SOLUTION - Question 5 Find the distance between the points (a, b) and (−a, −b). VIEW SOLUTION - Question 6 Let ∆ ABC ∽ ∆ DEF and their areas be respectively, 64 cm2 and 121 cm2. If EF = 15⋅4 cm, find BC. VIEW SOLUTION - Question 7 Find the solution of the pair of equation : Find the value(s) of k for which the pair of equations has a unique solution. VIEW SOLUTION - Question 8 Use Euclid's division algorithm to find the HCF of 255 and 867. VIEW SOLUTION - Question 9 The point R divides the line segment AB, where A(−4, 0) and B(0, 6) such that Find the coordinates of R. VIEW SOLUTION - Question 10 How many multiples of 4 lie between 10 and 205 ? Determine the A.P. whose third term is 16 and 7th term exceeds the 5th by 12. VIEW SOLUTION - Question 11 Three different coins are tossed simultaneously. Find the probability of getting exactly one head. VIEW SOLUTION - Question 12 A die is thrown once. Find the probability of getting. (a) a prime number. (b) an odd number VIEW SOLUTION - Question 13 In Figure 1, BL and CM are medians of a ∆ABC right-angled at A. Prove that 4 (BL2 + CM2) = 5 BC2.OR Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals. VIEW SOLUTION - Question 14 In Figure 2, two concentric circles with centre O, have radii 21 cm and 42 cm. If ∠AOB = 60°, find the area of the shaded region. - Question 15 A cone of height 24 cm and radius of base 6 cm is made up of modelling clay. A child reshapes it in the form of a sphere. Find the radius of the sphere and hence find the surface area of this sphere. A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank in his field which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of 3 km/hr, how much time will the tank be filled ? VIEW SOLUTION - Question 16 Calculate the mode of the following distribution : Class : 10 − 15 15 − 20 20 − 25 25 − 30 30 − 35 Frequency : 4 7 20 8 1 - Question 17 Show that is not a rational number, given that is an irrational number. VIEW SOLUTION - Question 18 Obtain all the zeroes of the polynomial 2x4 − 5x3 − 11x2 + 20x + 12 when 2 and − 2 are two zeroes of the above polynomial VIEW SOLUTION - Question 19 A motorboat whose speed is 18 km/hr in still water takes on hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream. VIEW SOLUTION - Question 20 (sin θ + 1 + cos θ) (sin θ − 1 + cos θ) . sec θ cosec θ = 2 Prove that : - Question 21 In what ratio does the point P(−4, y) divide the line segment joining the points A(−6, 10) and B(3, −8) ? Hence find the value of y. Find the value of p for which the points (−5, 1), (1, p) and (4, −2) are collinear. VIEW SOLUTION - Question 22 ABC is a right triangle in which ∠B = 90°. If AB = 8 cm and BC = 6 cm, find the diameter of the circle inscribed in the triangle. VIEW SOLUTION - Question 23 In an A.P., the first term is −4, the last term is 29 and the sum of all its terms is 150. Find its common difference VIEW SOLUTION - Question 24 Draw a circle of radius 4 cm. From a point 6 cm away from its centre, construct a pair of tangents to the circle and measure their lengths VIEW SOLUTION - Question 25 Prove that : 2(sin6 θ + cos6 θ) − 3 (sin4 θ + cos4 θ) + 1 = 0 VIEW SOLUTION - Question 26 Solve for x : The sum of the areas of two squares is 640 m2. If the difference of their perimeters is 64 m, find the sides of the square. VIEW SOLUTION - Question 27 In ∆ ABC (Figure 3), AD ⊥ BC. Prove that AC2 = AB2 +BC2 − 2BC × BD - Question 28 A moving boat is observed from the top of a 150 m high cliff moving away from the cliff. The angle of depression of the boat changes from 60° to 45° in 2 minutes. Find the speed of the boat in m/min. There are two poles, one each on either bank of a river just opposite to each other. One pole is 60 m high. From the top of this pole, the angle of depression of the top and foot of the other pole are 30° and 60° respectively. Find the width of the river and height of the other pole. VIEW SOLUTION - Question 29 Calculate the mean of the following frequency distribution : Class : 10−30 30−50 50−70 70−90 90−110 110−130 Frequency : 5 8 12 20 3 2 The following table gives production yield in kg per hectare of wheat of 100 farms of a village : 40−45 45−50 50−55 55−60 60−65 65−70 Number of farms 4 6 16 20 30 24 Change the distribution to a 'more than type' distribution, and draw its ogive. VIEW SOLUTION - Question 30 A container opened at the top and made up of a metal sheet, is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost of milk which can completely fill the container, at the rate of ₹ 50 per litre. Also find the cost of metal sheet used to make the container, if it costs ₹ 10 per 100 cm2. (Take π = 3⋅14) VIEW SOLUTION

- Integral element In commutative algebra, an element b of a commutative ring B is said to be integral over its subring A if there are such that If A, B are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since one does not have to insist on "monic".) The special case of greatest interest in number theory is that of complex numbers integral over Z; in this context, they are usually called algebraic integers (e.g., .) A ring consists of some (not all) algebraic integers is called the ring of integers, a central object in algebraic number theory. - Integers are the only elements of Q that are integral over Z (Thus, Z is the integral closure of Z in Q.) - Gaussian integers, complex numbers of the form , are integral over Z. (cf. quadratic integers.) is then the integral closure of Z in . - The roots of unity and nilpotent elements are integral over Z. - The integral closure of Z in the field of complex numbers C is called the ring of algebraic integers. - Let a finite group G act on a ring A. Then A is integral over AG the set of elements fixed by G. - Let R be a ring and u a unit in a ring containing R. Then (i) u − 1 is integral over R if and only if (ii) is integral over R. Let B be a ring, and let A be a subring of B. Given an element b in B, the following conditions are equivalent: - (i) b is integral over A; - (ii) the subring A[b] of B generated by A and b is a finitely generated A-module; - (iii) there exists a subring C of B containing A[b] and which is a finitely-generated A-module; - (iv) there exists a finitely generated A-submodule M of B with and the annihilator of M in B is zero. - Theorem Let u be an endmorphism of an A-module M generated by n elements and I an ideal of A such that . Then there is a relation: This theorem (with I = A and u multiplication by b) gives (iv) (i) and the rest is easy. (Note the generality on an ideal I is useful for the consideration of the integral closure of an ideal.) Coincidentally, Nakayama's lemma is also an immediate consequence of this theorem. It follows from the above that the set of that is integral over A forms a subring of B containing A. It is called the integral closure of A in B. The proof is due to Dedekind (Milne, ANT). Alternatively, one can use symmetric polynomials to show integral elements form a ring. (loc cit.) If A happens to be the integral closure of A, then A is said to be integrally closed in B. If A is reduced (e.g., a domain) and B its total ring of fractions, one often drops qualification "in B" and simply says "integral closure" and "integrally closed." Similarly, "integrality" is transitive. Let C be a ring containing B and c in C. If c is integral over B and B integral over A, then c is integral over A. In particular, if C is itself integral over B and B is integral over A, then C is also integral over A. If A is noetherian, one has a simpler criterion for integrality: b is integral over A if and only if there is a nonzero d such that for all . This can be used to weaken (iii) in the above to - (iii) bis There exists a finitely generated A-submodule of B that contains A[b]. Finally, the assumption that A be a subring of B can be modified a bit. If f: A B is a ring homomorphism, then one says f is integral if f(A) is integral over B, in the same way one says f is finite (B finitely generated A-module) or of finite type (B finitely generated A-algebra). In this view point, one can says - f is finite if and only if f is integral and of finite-type. Or, more explicitly, - B is a finitely generated A-module if and only if B is generated as A-algebra by a finite number of elements integral over A. One of the Cohen-Seidenberg theorems shows that there is a close relationship between the prime ideals of A and the prime ideals of B. Specifically, they show that an integral extension A⊆B has the going-up property, the lying over property, and the incomparability property. In particular, the Krull dimension of A and B are the same. When A, B are domains, A is a field if and only if B is a field. Let be an integral extension of rings. Then the induced map is closed. This is a geometric interpretation of the going-up property. Let be rings and A' the integral closure of A in B. (See above for the definition.) Integral closures behave nicely under various construction. Specifically, the localization S−1A' is the integral closure of S−1A in S−1B, and A'[t] is the integral closure of A[t] in B[t]. The integral closure of a local ring A in, say, B, need not be local. This is the case for example when A is Henselian and B is a field extension of the field of fractions of A. If A is a subring of a field K (A is necessarily a domain), then the integral closure of A in K is the intersection of all valuation rings of K containing A. Assume A is reduced. The conductor of A is : it is the largest ideal of A that is also an ideal of A'. If the conductor is A, then A is integrally closed. Note this is a generalization of the same concept in algebraic number theory. There is a concept of the integral closure of an ideal. The integral closure of an ideal , usually denoted by , is the set of all elements such that there exists a monic polynomial with with r as a root. The integral closure of an ideal is easily seen to be in the radical of this ideal. There are alternate definitions as well. - if there exists a not contained in any minimal prime, such that for all sufficiently large n. - if in the normalized blow-up of I, the pull back of r is contained in the inverse image of I. The blow-up of an ideal is an operation of schemes which replaces the given ideal with a principal ideal. The normalization of a scheme is simply the scheme corresponding to the integral closure of all of its rings. The notion of integral closure of an ideal is used in some proofs of the going-down theorem. Noether's normalization lemma Noether's normalisation lemma is a theorem in commutative algebra. Given a field K and a finitely generated K-algebra A, the theorem says it is possible to find elements y1, y2, ..., ym in A that are algebraically independent over K such that A is finite (and hence integral) over B = K[y1,..., ym]. Thus the extension K ⊂ A can be written as a composite K ⊂ B ⊂ A where K ⊂ B is a purely transcendental extension and B ⊂ A is finite. - ^ The above equation is sometimes called an integral equation and b is said to be integrally dependent on A (as opposed to algebraic dependent.) - ^ Kaplansky, 1.2. Exercise 4. - ^ Chapter 2 of Huneke and Swanson 2006 - ^ An exercise in Atiyah-MacDonald. - ^ Chapter 12 of Huneke and Swanson 2006 - ^ Chapter 4 of Reid. - M. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison–Wesley, 1994. ISBN 0201407515 - Kaplansky, Irving (September 1974). Commutative Rings. Lectures in Mathematics. University of Chicago Press. ISBN 0226424545. - H. Matsumura Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8. - J. S. Milne, "Algebraic number theory." available at http://www.jmilne.org/math/ - Huneke, Craig; Swanson, Irena (2006), Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, 336, Cambridge, UK: Cambridge University Press, ISBN 978-0-521-68860-4, MR2266432, http://people.reed.edu/~iswanson/book/index.html - M. Reid, Undergraduate Commutative Algebra, London Mathematical Society, 29, Cambridge University Press, 1995. 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What are dBm?March 22, 2023 RF signal power is often expressed in a logarithmic format. dBm refers to the ratio of the RF power to 1 milliwatt in logarithmic format. For instance, 1 Watt is 1000 milliwatts, so the ratio is 1000 and Log10(1000) gives 30 dBm. In general: The RMS voltage of an RF signal in a characteristic impedance R is given by: Some RMS and peak voltages in a 50Ω system are shown in the table below. Table – RMS and Pk RF Voltage vs dBm Notably, the peak RF voltages at +10 dBm and +30 dBm are 1V and 10V respectively. This is useful to know when evaluating low frequency signal levels with an oscilloscope. See this and more important radio frequency charts and formulas in this comprehensive application note. Can I Use a VNA as a Signal Generator? September 12, 2023 Yes! We often are asked about using a vector network analyzer for another purpose, as test equipment budgets may need to stretch. But the answer to this question is yes, you can use a VNA as an additional signal source for testing. Below are some basics of how to do it but there is a more robust application note available here to review for more details.You can set the sweep range to “Zero Span” and set the frequency to the desired value and the VNA will output a constant frequency tone. However, there are some caveats. The Compact Series from CMT uses fractional-N phase locked loops to generate signals with high resolution, thus the output frequency may be off by a fraction of a hertz. Therefore, if the vector network analyzer is used as a frequency source, another signal generator is used to generate that same frequency, and the two instruments are provided with the same 10 MHz time-base reference on the rear panel, the VNA signal may not be frequency locked to the generator. However, this would be an unusual requirement.It is generally not necessary for a VNA to hit frequencies precisely. It is only important that the measured internal IF frequencies fall within the IF bandwidth of the DSP filter implemented in the FPGA.The Cobalt Series VNAs, either 9 or 20 GHz use Direct Digital Synthesis (DDS) to generate the fine frequency steps so in that case the output frequencies will be exact.How To Use a VNA as a Signal GeneratorThe simplest way to configure the analyzer as a signal generator is simply to issue a Preset (System > Preset using the menus along the right side of the user interface), then set the Span to 0 Hz, and finally set the Center frequency and output power to the desired signal source settings.To change the output port of the signal from Port 1 to Port 2, simply change the measured S-parameter from S11 to S22. If you need not retain your calibration and display settings, that’s all there is to it! If you do need to keep such settings, however, the sections that follow walk through the steps necessary to generate a CW starting from typical instrument S-parameter sweep settings without a Preset.When using the VNA as a signal source, it’s important to keep in mind certain limitations which might affect its suitability depending on your application:The generator of the VNA will exhibit harmonics of the fundamental frequency as high as its specified harmonic distortion. You can find the specified harmonic distortion of each CMT VNA in its corresponding datasheet; typically, those are approximately -25 dBc.There will also be non-harmonic distortion, typically lower in power than the harmonic distortion.The generated signal’s output power will be subject to the output power accuracy specification of the instrument; +/- 1.0 dB is a typical specification. For more precise output powers, a power meter should be used to confirm or adjust the VNA output power.Some output power levels for CMT’s 1-Port VNAs are High/Low, specified as “Typical Only” and can’t be adjusted. Output power will need to be set with external attenuators or amplifiers according to the application at hand.If sweeping is halted by setting the trigger mode to HOLD, don’t forget that changes made to the output power or frequency will not take effect until a sweep is initiated! You can click on “Single” to apply the new settings and automatically return to HOLD mode after the sweep completes.For those engineers curious about additional uses for a vector network analyzer, CMT has additional resources to explain the options and pros and cons for each. For example, spectrum analyzer vs network analyzer; what are the differences and can a VNA be used as a spectrum analyzer is expanded in this application note. What Are the VNA Display Formats and What Do They Mean? August 10, 2023 Introduction Fundamentally, a Vector Network Analyzer (VNA) measures RF reflections. In a 50Ω system, reflections occur whenever an impedance other than 50 ohms is encountered by the RF signal. The reflection has a magnitude and a phase with respect to the magnitude and phase of the incident signal and may be characterized as a complex number. The VNA menu allows for a selection of charts which can display complex reflection coefficients in many different ways. Read on to learn more about VNA display formats available for analyzing measurements. Reflection and Reflection Coefficients In optics, if a beam of light traveling in a media with dielectric constant ε1 encounters a region with dielectric constant ε2, there will be a reflection as shown in Figure 1. Figure 1 - Optical Reflection The magnitude of the reflection is related to the difference between the two dielectric constants. The transmitted and reflected waves may be normalized by dividing by the incident wave to arrive at a Reflection Coefficient, Γ. For instance, in the absence of loss, the coefficient for reflection might be 0.2, thus the coefficient for transmission must be 0.8. Here, the reflection coefficient is ratio-metric, and the magnitude of the incident signal is irrelevant. Similarly, an RF voltage wave travelling on a 50Ω transmission line may encounter an impedance other than 50 ohms. This results in a reflection. Specifically, that reflection is given by: Where Z is the encountered impedance. If Z is a complex impedance –which can be expressed as R+jX– such as a resistor and capacitor in series, the reflection coefficient Γ will also be complex. We can choose to display the complex reflection coefficient in the following ways: VNA Display Formats Log Magnitude In this format, the magnitude of the complex reflection coefficient is calculated, the base 10 logarithm is applied, and the result multiplied by 20. The multiplication by 20 makes the result proportional to power instead of voltage. Where the complex reflection coefficient Γ is a+jb. This format is convenient to examine a reflection coefficient over a very wide dynamic range. Details of the low loss in the passband of a filter along with a very deep stopband can be clearly seen in Figure 2. This would not be possible using a linear scale. Figure 2 - Filter Plot Linear Magnitude The linear scale is convenient to examine a transmission coefficient such as S21, where the linear magnitude covers a range within two to three orders of magnitude. Figure 3 shows the same filter from Figure 2 but in the linear scale. Details at markers 2 and 3 are not distinguishable in this format. Figure 3 - Filter in Linear Scale Delay The delay format shows the group delay, which is calculated from -dФ/dω of S21. Ф is the phase of the reflection coefficient and ω is the radian frequency, 2πf. The group delay of the filter in Figure 2 is shown below in Figure 4. Figure 4 - Filter Group Delay VNA Display Format The group delay varies from 7.2 nS to 15.4 nS in the passband of the filter. If a modulated signal were passed through this filter such that some of the modulation bandwidth extends toward the peaks, there would be significant phase distortion (dispersion). Frequencies near the peaks might arrive at the receiver 5 nS or so later than frequencies near the center. Filters can be designed with much flatter delay than this if needed, and this can be evaluated using this delay display mode. Phase The phase of the reflection coefficient can be displayed on the VNA screen. The phase is the arctangent of the ratio of the imaginary and real parts of the coefficient, but it represents the phase alignment of the reflected or transmitted signal with respect to the incident signal. If the two signals are in perfect alignment, the phase difference is zero. The phase depends greatly on the delay experienced by the signal. Figure 5 shows the phase of the signal passed through the filter of Figure 2. Figure 5 - Filter S21 Phase The phase wraps at -180 and 180 degrees. The curvature on the left and right sides is due to the delay variation shown in Figure 4, whereas the phase is reasonably linear where the delay is flat in the middle of the passband. Extended Phase The phase is not wrapped in extended phase mode. The phase will be near zero at low frequencies and then become more and more negative at higher frequencies. The phase will change at a greater rate as more delay is experienced. Figure 6 - Extended Phase Display SWR The Standing Wave Ratio is used on reflection measurements such as S11. Reflections on a transmission line interfere with the incident wave, creating peaks and valleys in the voltage envelope of the RF wave. The ratio of the peaks to the valleys along the transmission line is the SWR. An SWR of 1:1 means there are no peaks or valleys, and hence no reflection. SWR may be calculated from the linear reflection coefficient, Γ. Real and Imag The Real and Imaginary display formats depict the real or the imaginary components of a reflection coefficient. Note, these are not the real and imaginary components of an impedance. If there is no imaginary component, then the impedance contains no reactive component and must be resistive alone. If there is no real component, the impedance must be reactive alone. Polar The polar format plots the complex reflection coefficient within a unit circle with the real component on the horizontal axis and the imaginary component on the vertical axis. This is the most fundamental plot, as it displays the reflection coefficient in its most basic form. All other formats are useful derivations. A reflection coefficient is essentially a complex vector. The polar format plots the magnitude and phase of this vector. A vector is often drawn as an arrow, but a dot at what would be the tip of the vector is drawn to reduce clutter when showing vectors for many frequencies. A VNA display format of a polar chart will have circles of magnitude and radial lines of angle, as shown below in Figure 7. Magnitude is shown in linear format here. Figure 7 - Polar Plot If Polar(log) is chosen, the magnitudes of the makers will be stated in dB format. The polar Smith formats, Smith(log), Smith(Lin), Smith(Re/Im), and Smith(R+jX) are Polar plots with a Smith chart overlay. The Smith overlay maps the polar reflection coefficient to the corresponding impedance, which gives rise to it as calculated from Eq. 1. The variations above merely change the values given by the markers. Figure 8 - Smith(R+jX) Format Smith(log) will give the log magnitude and angle of the reflection. Smith(Lin) will give the linear magnitude and angle. Smith(Re/Im) will give the linear Cartesian format reflection. Smith(R+jX) will give the complex impedance at each point. Smith(G+jB) uses the admittance overlay on the polar plot. Instead of an R+jX impedance, the marker value will show the G+jB admittance, which gives rise to the reflection, where the R+jX mode gives the impedance of a pair of series components, a resistance and a reactance. The equivalent inductance or capacitance at that frequency is also given. The G+jB admittance gives the conductance (inverse of resistance) and the susceptance of two components in parallel. The Smith(R+jX) is convenient for analyzing an impedance, which is best modeled as a series impedance, while Smith(G+jB) is best for analyzing a parallel admittance. Conclusion Different display formats are useful for different measurements. This application note covers several different display formats and their uses. For example, the Log Mag format might be best for evaluating a filter, but the SWR will likely be used to evaluate the input match of an antenna, and the Smith(R+jX) is useful for measuring an impedance and perhaps working out a matching network. Useful Radio Frequency Engineering Formulas and Charts February 9, 2023 This application note is a collection of essential formulas and charts for Radio Frequency Engineering. Using a VNA as a Signal Generator April 19, 2018 A common question from users of Copper Mountain Technologies’ USB-based Vector Network Analyzers is whether the analyzer can be used as a signal source. Users’ motivations for doing so vary, but most commonly the question arises when an additional source is needed in a test setup but is not available. For example, a source might be needed as the LO to a mixer, to check functionality of another test equipment like a power meter or spectrum analyzer, or to produce a reference clock for use elsewhere in a test system. Fortunately, any CMT VNA can readily be used as a signal source. This application note describes the process of configuring a CMT VNA as a signal source, and the expected performance of an analyzer so-configured. This application note is based on the S2 family of instruments’ software; other instruments will follow similar menu structures and procedures.

Cambridge University Press. The null hypothesis is "defendant is not guilty;" the alternate is "defendant is guilty."4 A Type I error would correspond to convicting an innocent person; a Type II error would correspond A Type II error is a false NEGATIVE; and N has two vertical lines. Credit has been given as Mr. http://u2commerce.com/type-1/type-1-versus-type-2-error.html Reply ATUL YADAV says: July 7, 2014 at 8:56 am Great explanation !!! Raiffa, H., Decision Analysis: Introductory Lectures on Choices Under Uncertainty, Addison–Wesley, (Reading), 1968. I think your information helps clarify these two "confusing" terms. Medical testing False negatives and false positives are significant issues in medical testing. As a result of the high false positive rate in the US, as many as 90–95% of women who get a positive mammogram do not have the condition. Read More Share this Story Shares Shares Send to Friend Email this Article to a Friend required invalid Send To required invalid Your Email required invalid Your Name Thought you might Cambridge University Press. Collingwood, Victoria, Australia: CSIRO Publishing. While most anti-spam tactics can block or filter a high percentage of unwanted emails, doing so without creating significant false-positive results is a much more demanding task. So a "false positive" and a "false negative" are obviously opposite types of errors. share|improve this answer answered Nov 3 '11 at 1:20 Kara 311 add a comment| up vote 3 down vote I am surprised that noone has suggested the 'art/baf' mnemonic. Type 1 Error Psychology p.28. ^ Pearson, E.S.; Neyman, J. (1967) . "On the Problem of Two Samples". The rate of the typeII error is denoted by the Greek letter β (beta) and related to the power of a test (which equals 1−β). Power Of The Test For example, all blood tests for a disease will falsely detect the disease in some proportion of people who don't have it, and will fail to detect the disease in some The boy's cry was alternate hypothesis because a null hypothesis is no wolf ;) share|improve this answer edited Mar 24 '12 at 23:51 naught101 1,8402554 answered Oct 21 '11 at 21:49 They also cause women unneeded anxiety. ISBN1-599-94375-1. ^ a b Shermer, Michael (2002). read this post here Thanks. –forecaster Dec 28 '14 at 20:54 add a comment| up vote 9 down vote I'll try not to be redundant with other responses (although it seems a little bit what Probability Of Type 1 Error The null hypothesis is that the input does identify someone in the searched list of people, so: the probability of typeI errors is called the "false reject rate" (FRR) or false Type 3 Error One consequence of the high false positive rate in the US is that, in any 10-year period, half of the American women screened receive a false positive mammogram. Biometrics Biometric matching, such as for fingerprint recognition, facial recognition or iris recognition, is susceptible to typeI and typeII errors. news The Skeptic Encyclopedia of Pseudoscience 2 volume set. Or in other-words saying that it the person was really innocent there was only a 5% chance that he would appear this guilty. You can also subscribe without commenting. 22 thoughts on “Understanding Type I and Type II Errors” Tim Waters says: September 16, 2013 at 2:37 pm Very thorough. Type 1 Error Calculator Statistical calculations tell us whether or not we should reject the null hypothesis.In an ideal world we would always reject the null hypothesis when it is false, and we would not The blue (leftmost) curve is the sampling distribution assuming the null hypothesis ""µ = 0." The green (rightmost) curve is the sampling distribution assuming the specific alternate hypothesis "µ =1". Statistical significance The extent to which the test in question shows that the "speculated hypothesis" has (or has not) been nullified is called its significance level; and the higher the significance have a peek at these guys A type II error would occur if we accepted that the drug had no effect on a disease, but in reality it did.The probability of a type II error is given For example, when examining the effectiveness of a drug, the null hypothesis would be that the drug has no effect on a disease.After formulating the null hypothesis and choosing a level Types Of Errors In Accounting ISBN1584884401. ^ Peck, Roxy and Jay L. Sign in to add this to Watch Later Add to Loading playlists... Suggestions: Your feedback is important to us. Sometimes different stakeholders have different interests that compete (e.g., in the second example above, the developers of Drug 2 might prefer to have a smaller significance level.) See http://core.ecu.edu/psyc/wuenschk/StatHelp/Type-I-II-Errors.htm for more But the increase in lifespan is at most three days, with average increase less than 24 hours, and with poor quality of life during the period of extended life. Types Of Errors In Measurement ABC-CLIO. Launch The “Thinking” Part of “Thinking Like A Data Scientist” Launch Determining the Economic Value of Data Launch The Big Data Intellectual Capital Rubik’s Cube Launch Analytic Insights Module from Dell If you could test all cars under all conditions, you would see an increase in mileage in the cars with the fuel additive. Plus I like your examples. http://u2commerce.com/type-1/type-1-and-type-2-error-statistics-examples.html She said that during the last two presidencies Republicans have committed both errors: President ONE was Bush who commited a type ONE error by saying there were weapons of mass destruction Freddy the Pig View Public Profile Find all posts by Freddy the Pig #16 04-17-2012, 11:33 AM GoodOmens Guest Join Date: Dec 2007 In the past I've used Null Hypothesis Type I Error / False Positive Type II Error / False Negative Wolf is not present Shepherd thinks wolf is present (shepherd cries wolf) when no wolf is actually Email Address Please enter a valid email address. In Type I errors, the evidence points strongly toward the alternative hypothesis, but the evidence is wrong. Last updated May 12, 2011 Straight Dope Message Board > Main > General Questions Type I vs Type II error: can someone dumb this down for me User Contact Us - Straight Dope Homepage - Archive - Top Powered by vBulletin Version 3.8.7Copyright ©2000 - 2016, vBulletin Solutions, Inc. We say, well, there's less than a 1% chance of that happening given that the null hypothesis is true. Although they display a high rate of false positives, the screening tests are considered valuable because they greatly increase the likelihood of detecting these disorders at a far earlier stage.[Note 1]

Sal subtracts 14 - 6 by first thinking about subtracting 2 and 4. Created by Sal Khan. Want to join the conversation? - I am struggling to find a video on Subtracting within 20 with regrouping. I dont want to use blocks, I don't want to use visualizations. I want to know how to know how to write subtract 9 from 16 with regrouping.(7 votes) - Try this, We have the original number, but we can put it in expanded form*. So, 16 = 10 + 6, and we can use this. We know that 10 - 9 = 1, and then we can add the remaining 6 to get 7. Expanded form is just taking the values of each number and adding them to create a number. For example, 17 = 10 + 7 You can also use this for bigger numbers, like 1,234 = 1,000 + 200 + 30 + 4(18 votes) - Hi there, Could you please tell which video explains how to do 9-=5 and -4=5. Any help on this is much appreciated.(5 votes) In these type of problems all you have to do is do the opposite of what it's asking. But, you would, in this case, still subtract since you are trying to find the number you subtract by and not the number you subtract from. So, 9-5=4 the missing value is 4. For the second problem you add since you are trying to find the number you subtract from. So, 4+5=9 the missing value is 9. Hope this helps!(1 vote) - What is the commutative property? Is addition commutative?(1 vote) - It is when you can change the order of an equation and get the same answer. For Ex: 4+3= 7 is the same as 3+4=7(0 votes) - If your subtracting by adding a negative number does that lead to borrowing to significant to basic subtraction?(0 votes) - The basic concept of adding a negative number is exactly the same as subtracting that number. In higher math this may not always be the case but you will learn about those types of situations later. For now think about adding a negative number as the same process as subtraction. For example 10 + -3 = 7 and 10 - 3 = 7.(0 votes) - so people can think of the four's cancelling each other out in the second problem and being left with only ten?(0 votes) - Yes, but not all math problems are simple like that. But it is definitely not encouraged to think as such, as people might think that 14 - 4 = 1 if they cancel out the 4 and forget that there is a 0 behind the 1.(0 votes) - Why is subtraction not commutative?(0 votes) - An operation is commutative if you can change the order and get the same answer. 2 - 1 = 1 1 - 2 = -1 Therefore, subtraction is not commutative.(0 votes) Voiceover:Let's see if we can compute what 14 - 2 is, and then figure out what 14 - 4 is, and then figure out what 14 - 6 is. And I encourage you to pause this video and try to figure these out before I work through them. Voiceover:So I'm assuming you've given it a try, now let's think about it. The number 14 just by how it's written, we know that it's going to be 1 group of ten. That's what this 1 tells us plus another 4 ones. So let's verify that we have 14 objects down here. This is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. So we definitely have a group of ten. We have 1 group of ten here, so that's my group of ten. So let me box that off, so I have my group of ten. And then how many ones do I have? I have 1, 2, 3, 4, I have four ones. So this is indeed, 14. This one that we have right over here, this represents this ten, it's 1 group of ten. This is 1 group of ten, let me write that down. Group of ten. And then we have 4 ones, 4, I guess we could say, 4 ones like that, so that is indeed 14. Now let's look at each of these, what is 14-2 going to be? Well we could take away 2, so take away 1... take away 2, and how many are we left with? Well we still have our one group of ten, so it's going to be 1, followed by how many ones do we have left over, well we have 2 ones. We have 2 ones left over. 1, 2, so we are going to be left with 12. Now what about 14 - 4? So let me clear this out. Actually let me do it like this. So if I clear that, so what is 14 - 4 going to be? So now I'm going take away 1, 2, 3, and 4. I have essentially taken away all 4 ones. So what am I left with? Well I still have my 1 group of ten, I still have my 1 group of ten and I have 0 ones left over. No ones anymore, so now I have 0 ones. And so 14 - 4 is 10, and that makes sense. 14 is 10 + 4 and now we're subtracting 4, to get 10, let me write this down. This is the same thing, 14 is 10 + 4. Voiceover:10 + 4, this is 14 and then we're going to subtract 4, Then we're subtracting 4, so if you have 10 + 4 - 4. Well the 4 - 4 is going to be 0, you're going to be left with 10 + 0 or just 10. We can do the same thing up here. This is equal to 14 is 10 + 4 and then we subtracted 2. And then we subtracted 2. So what you're going to be left with 10 + what's 4 - 2? It's 2, so that's what we got right over here. Let me make this very clear, this simplifies to 2. This 2 right over here. This simplifies to 0, this 0 right over here. Now let's do the last one, what is 14 - 6? Well we're going to take away 1, 2, 3, 4, then 5, and 6, so now we've broken into our group of ten. So this is going to be a one-digit number. And we are left with 1, 2, 3, 4, 5, 6, 7, 8. So this is going to be equal to 8 and we're all done.

A brief introduction of color ring resistance identification method Color ring resistance identification method refers to the resistance above the use of four color rings or five color rings or six color rings to represent the resistance value. The color information representing the resistance value can be read from any Angle at once. Color ring marking is mainly used on cylindrical resistors, such as carbon film resistors, metal film resistors, metal oxide film resistors, fuse resistors, winding resistors. Second, color ring resistance identification sequence Color ring resistance is the most common type of resistance used in various electronic equipment. No matter how it is installed, the repairman can easily read its resistance value for easy detection and replacement. However, in practice, it is found that the sequence of some color ring resistors is not very clear, which is often easy to read wrong. In recognition, the following skills can be used to judge: Technique 1: first find the color ring of the error mark, so as to arrange the color ring order. The most commonly used colors for resistance error are: gold, silver, and brown, especially gold and silver rings, which are rarely used as the first ring of resistance color ring, so as long as there are gold and silver rings on the resistance, you can basically identify this as the last ring of resistance color ring. Technique 2: the identification of whether the brown ring is an error mark. Brown rings are often used as both error rings and significant number rings, and often appear in both the first and last rings, making it difficult to identify the first ring. In practice, it is possible to judge by the spacing between the color rings: for example, for the resistance of a five-channel color ring, the spacing between the fifth and fourth rings is wider than that between the first and second rings, thus determining the order of the color rings. Tip 3: In the case that the color ring order cannot be determined only by the color ring spacing, the production sequence value of the resistance can also be used to distinguish. For example, there is a resistance color ring read sequence: brown, black, black, yellow, brown, its value is: 100×10000=1M ω error 1%, belongs to the normal resistance series value, if the reverse order read: brown, yellow, black, black, brown, its value is 140×1 ω =140 ω, error 1%. Obviously, the resistance value read in the latter order is not available in the production series of resistors, so the latter color ring order is not correct. This picture is provided by registered user “Internet said”, feedback of copyright statement Three, color ring resistance identification method In the early days, color ring marking was used to indicate resistance values, tolerances and specifications of resistors when the surface of resistors was not sufficient for numerical representation. There are two main parts. The first part: the group near the front of the resistor is used to indicate the resistance value. The resistance value of two significant numbers is represented by the first three color rings, such as 39 ω, 39K ω, and 39M ω. The resistance value of the three effective numbers is represented by the four color rings, such as 69.8 ω, 698 ω, and 69.8K ω. It is generally used for the expression of precision resistance. Part two: A color ring near the rear end of the resistor is used to represent tolerance accuracy. Each color ring in Part 1 is isometric, self-contained and easily distinguishable from the color ring in Part 2. Tricolor ring resistor The first color ring is the ten digit number, the second color ring is the digit number, and the third color ring represents the multiplier. Use the first three color rings to represent the resistance value, such as 39 ω, 39K ω, 39M ω. Four color ring resistor Four color ring resistance identification: the first and second ring respectively represent two effective number of resistance value; The third ring represents multiplication; The fourth ring represents the error. Brown red gold Its resistance value is 12×10^2=1.2 K ω error is ±5% The error represents the resistance value, and the fluctuation of the standard value 1200 (5%×1200) indicates that the resistance is acceptable, that is, it is a good resistance between 1140-1260. The first and second rings with four color rings represent the first two digits of resistance value respectively; The third ring represents multiplication; The fourth ring represents the error. The key to rapid identification is to determine the resistance value in a certain order of magnitude range according to the color of the third ring, such as a few K, or dozens of several K, and then “substitute” the number read by the first two rings, so that the number can be quickly read. Five-color ring resistor Identification of five color ring resistance: the first, second and third rings respectively represent the resistance value of three effective numbers; The fourth ring represents multiplication; The fifth ring represents the error. If the fifth color ring is black, it is generally used as a winding resistor, and if the fifth color ring is white, it is generally used as a fuse resistor. If the resistance body has only a black color ring in the middle, it means that the resistance is zero ohm resistance. Example: red red black brown gold Its resistance is 220×10^1=2.2K ω and the error is ±5% The first color ring is hundreds, the second color ring is tens, The third color ring is the digit number, the fourth color ring is the power of color times the color, and the fifth color ring is the error rate. First, from the bottom of the resistor, find the color ring that represents tolerance accuracy, with gold representing 5% and silver representing 10%. In the above example, the end of the color ring is golden, so the error rate is 5%. Then from the other end of the resistance, find the first and second color rings and read their corresponding numbers. In the above example, the first three color rings are red, red and black, so their corresponding numbers are red 2, red 2 and black 0, and their effective number is 220. Read the fourth multiple color ring, brown 1. So, we get a resistance of 220 x 10^1=2.2K ω. That is, the resistance value between 2090-2310 is a good resistance. If the fourth multiple color ring is gold, multiply the significant number by 0.1. If the fourth multiple color ring is silver, multiply by 0.01. Six color ring resistor Six color ring resistance identification: six color ring resistance front five color ring and five color ring resistance is the same method, the sixth color ring represents the temperature coefficient of the resistance. Key points of color ring resistance recognition Take the four-color ring as an example: (1) Memorize the number represented by each color of the first and second rings. Brown 1, red 2, orange 3, yellow 4, green 5, blue 6, purple 7, gray 8, white 9, black 0. Read it together and repeat it several times. Keep in mind the order of magnitude represented by the color of the third ring, which is the key to fast recognition. Concrete is: (2) In terms of order of magnitude, they can be roughly divided into three large classes, namely: gold, black, and brown are ohmic; Red, orange and yellow are in the kilo-scale; Green and blue are megohm. I’ll divide it up just so you can remember it. (3) When the second ring is black, the color of the third ring represents an integer, i.e., several, dozens, hundreds of K ω, etc., which is the special case of reading, so pay attention to it. For example, if the third ring is red, its resistance is the whole number of K ω. (The third or fourth ring of a 4-ring resistance that represents a multiple is followed by several zeros, or if it is a negative number, the decimal point of the significant number is moved to the left.) (4) Remember the error represented by the color of the fourth ring, namely: gold is 5%; Silver 10%; Colorless 20%. Example 1 When the four color rings are yellow, orange, red and gold, and the third ring is red and the resistance range is several K ω, the reading is 4.3 K ω according to the numbers “4” and “3” represented by yellow and orange respectively. The fourth ring is gold which means the error is 5%. Example 2 When the four color rings are brown, black, orange and gold, the third ring is orange and the second ring is black, the resistance value should be tens of K ω, according to the number “1” represented by brown, the reading is 10K ω. The fourth ring is gold, with a margin of error of 5%. In some cases, it is also possible to compare the two starting colors, because the first color of the calculation will not be gold, silver, or black. If these 3 colors are near the edge, you need to do the reverse calculation. There are two ways to mark the color of the color ring resistor, one is to use the 4-color ring marking method, the other is to use the 5-color ring marking method. The difference between the two is that the first two digits of the 4-color ring represent the significant digits of the resistance, while the first three digits of the 5-color ring represent the significant digits of the resistance. The second-to-last digit of the 2-color ring represents the multiplier of the significant digits of the resistance, and the last digit represents the error of the resistance. For 4-color ring resistance, its resistance value calculation method is as follows: Resistance value = (number of first color ring *10+ number of second color ring) * the multiplier represented by number of third color ring For the 5 color ring resistance, its resistance calculation method is as follows: Resistance value = (number 1 color ring *100+ number 2 color ring *10+ number 3 color ring) * number of times represented by number 4 color ring

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Calculus · Math Trusted teacher: VIRTUAL AND PARTICULAR TUTORIES ARE GIVEN IN THE CITY OF MEDELLÍN COLOMBIA OF: ALGEBRA, TRIGONOMETRY, GRAL MATHEMATICS, OPERATIONAL MATHEMATICS, CALCULATION, INTEGRAL, VARIOUS VARIABLES, MULTIPLE YEARS, YEARS WITH MORE THAN 20 YEARS OF EXPERIENCE AND EXCELLENT PEDAGOGY. THE OBJECTIVE IS TO BE ABLE TO MAKE MATH SOMETHING EASY AND SIMPLE. MY CELL PHONE NUMBER IS: 302-360,2844. THE VIRTUAL CLASSES ARE IN PLATFORM AND FROM A TABLE THE STUDENT SITS LIKE IN THE CLASSROOM. Calculus · Math · Trigonometry Calculus, Physics and Chemistry are the basic foundation of a STEM career. These courses must be mastered in order to progress through your degree. This class will specialize in Calculus, Physics, and Chemistry. As an Electrical Engineer at the University at Buffalo, I am well equipped with the knowledge of these topics to bring you success for the future. My goal is to work with you in order to prepare you for exams. You can learn Calculus, Physics, and Chemistry. Believe in yourself and you can achieve the success that you deserve Chemistry · Physics · Calculus Meet even more great teachers. Try online lessons with the following real-time online teachers: Dr S Iyer - Geneva, Switzerland$55 Trusted teacher: I am Dr. S Iyer- a tutor with over 17 years of teaching experience as of 2021 and several hundreds of students from all parts of the globe. I teach one-on-one online over Zoom/ Skype/ Google Meet/ Teams using a pen tablet and the screen-share feature. Depending on and tailored to the learning style of the student (visual, kinesthetic/ tactile, etc.) my teaching adopts a great deal of "Learn by Viewing" or "Learn by Doing" methodology that leverages the power of web-based interactive tools so as to build a solid foundational understanding. I have helped several students to do remarkably well in areas like Algebra, Geometry, Trigonometry, Precalculus, Calculus, Statistics. 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Alan - Taichung, Taiwan$47 Trusted teacher: I specialize in tutoring and preparing Chemistry, Physics, and Calculus AB+BC for the AP and also the IB exam. My goal is to improve problem solving ability of each of my students and also to increase their long term memory retention of learning materials. I assign homework after every session, schedule periodic review sessions and quizzes, and provide succinct progress reports. I have many satisfied students, many of which have gone on to attend top universities around the world. Calculus · Physics · Chemistry This course is made to prepare the private students for any mathematics exam types and levels for mainly baccalaureate level 1 and level 2. As well to prepare the student for the university with a very strong background in any field of mathematics. Calculus · Algebra · Trigonometry Assist students with homework, teaching them how to perform the calculations needed to complete their assignments. I am also teach students basic classroom skills such as note-taking, studying, and test-taking. Teach students various mathematical concepts, processes, and computations. write progress reports detailing individual student progress Statistics · Algebra · Calculus Do you hate math or do complex mathematical formulas make no sense to you? No worries, that is what I am for. I hated math when I was a teenager, considered it a boring subject of numbers. When I learnt mathematics, I discovered that it is a language of nature that most of school teachers fail to demonstrate. So, I am not only going to teach you mathematics, but will also make you love the subject. The class will be tailored to the need of a student. Algebra · Calculus · Math I hold extensive knowledge in classroom management and various teaching methods so I am able to reach learners of all backgrounds. I enjoy helping students to reach their academic goals and gain an understanding of mathematical concepts. 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Probability and Statistics: Probability problems infer conclusions about characteristics of hypothetical data taken from the population. Probability explains the mathematical chance that something might happen, and is used in numerous day- to-day applications. Calculus : We can find the (or global) minimum and maximum values of a function as well as the relative (or local) minimum and maximum values of a function. Differential Equation: Differential equations are applied to predict the world around us, and used in a wide variety of disciplines, from biology, economics, physics, chemistry and engineering. It has applications in exponential growth and decay model, Newton's law of cooling, and Circuit theory, e.t.c. I have teaching experience for more than 15 years. My students are from all corners of the globe, and of all ages. I also make them learn Mathematics in English, so that they can excel in future. 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Also, we manage classes for AP, IB, IGCSE and for Canadian Diploma in Mathematics Calculus · Algebra · Math Eli - Helsinki, Finland$24 Trusted teacher: My name is Eli and I am here to help you in math. If you are looking for a person, who can explain math problems in the simplest possible way or who can help you with your homework, do not hesitate to contact me! As a teacher, I have more than 6 years' experience and I give lessons of math, calculus, algebra, matrices and geometry. Before each session, I prepare appropriate materials, like theory, examples and exercises and we will work on them together, during our session. In my classes, you can ask any questions that you have and we do not finish the class before making everything clear! Now, I am a PhD student in electrical engineering at Aalto University and I have to use my mathematical knowledge every single day; therefore, I see the importance of learning math. * At the moment, I only teach online. For this purpose, I use my pen tablet and I share my virtual whiteboard. Therefore, students can see me and also what I am writing, at the same time. I look forward to meeting you! Calculus · Algebra · Math My name is Anum, I am graduated in Mathematics and currently doing a master's in Informatics. I am here to teach maths, there are some areas in which I would be more interested to teach and they are Mathematics - Analysis - Matrices - Statistics - Algebra -calculus. The medium of instruction would be English. The first meeting would be free, to discuss academic plans, which will follow in the future. So if you are having trouble with certain areas in mathematics? You can contact me, I would love to help. I can also assist students with homework, projects, test preparation, papers, research, and other academic tasks. Do Mathematics in the simplest and fascinating ways Statistics · Calculus · Algebra Trusted teacher: I`ve been teaching Mathematics for more than 10 years, from elementary levels to college, 8 years of that was at local universities as guest lecturer. I hold Masters degree in Math and working for a long time as tutoring taught me how to teach! My classes would be online and I share my screen and use light pen, so your screen is my white board! It feels like I`m there beside you and moreover, you can have note saved! Gre test preparation · Calculus · Math Trusted teacher: I taught Maths for ten years in both private preparatory schools, and secondary schools (in Surrey and Somerset.) I also successfully ran, for six years, my own private maths tuition business in Guildford. I have a First-Class Degree in Mathematics/Economics from the University of Surrey. Maths is a subject that, often, students of any age are simply afraid of. 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We've completed two Monday Math Madness contests. Last Friday Blinkdagger announced that Joshua Zucker, director at Julia Robinson Mathematics Festival, was randomly selected as the winner of the 2nd contest. Now it's my turn to post a contest problem. Those of you who are astute readers may have noticed that I said the contest would be held the 1st and 3rd Mondays of the month and today is actually the 5th Monday of March. Well, there's enough enthusiasm about this contest so we'll just do it every other week. So, we'll do 26 contests per year rather than 24. We're nice that way! Earlier this month UC Berkeley professor emeritus of mathematics David Gale passed away. Gale made a number of significant contributions to mathematics and he loved puzzles, games, and finding beauty in mathematics. Gale's daughter had this to say: On March 3rd Blinkdagger and I posted the first Monday Math Madness problem. On March 11th, after the first contest ended, I posted a couple of different solutions to the problem. Pat Ballew, even though he wasn't picked as the random winner, impressed me with a very clever solution to the problem that generalizes very nicely. He uses an approach called Markov state matrices, which I had never heard of. It seems to me that this approach is pretty similar to the one I posted from Richard Berlin. Pat and I exchanged several emails where he explained the method and here is my attempt to explain what Pat explained to me. This was the problem: A popular blog has just three categories: brilliant, insightful, and clever. Every blog post belongs to exactly one of the three categories and the category for each post is selected at random. What is the probability of reading at least one post from each category if a reader reads exactly five posts? Pat's approach starts by creating a matrix that encodes the probabilities of going from one "state" to another as a new blog post is read. State just refers to whether 0, 1, 2, or 3 categories have been encountered after reading some number of blog posts. After one blog post has been read we are in state 1 (1 category has been read). After two posts have been read we may be in state 1 (if both blog posts are in the same category), or state 2 (if the two categories are different), but not in state 3 (you could not have encountered three categories after having read only two blog posts.) Here's a joke I got a good chuckle out of. I'm not sure who to credit since there a number of web-sites with this joke so I'll credit the site where I first found it, Savage Research. Two mathematicians were having dinner in a restaurant, arguing about the average mathematical knowledge of the American public. One mathematician claimed that this average was woefully inadequate, the other maintained that it was surprisingly high. "I'll tell you what," said the cynic. "Ask that waitress a simple math question. If she gets it right, I'll pick up dinner. If not, you do." He then excused himself to visit the men's room, and the other called the waitress over. "When my friend comes back," he told her, "I'm going to ask you a question, and I want you to respond `one-third x cubed.' There's twenty bucks in it for you." She agreed. The cynic returned from the bathroom and called the waitress over. "The food was wonderful, thank you," the mathematician started. "Incidentally, do you know what the integral of x squared is?" The waitress looked pensive; almost pained. She looked around the room, at her feet, made gurgling noises, and finally said, "Um, one-third x cubed?" So the cynic paid the check. The waitress wheeled around, walked a few paces away, looked back at the two men, and muttered under her breath, "...plus a constant." Here are some interesting recent Math-related posts in the "blathosphere." You have only a couple more days to get your solutions in for the second Monday Math Madness at Blinkdagger. Blinkdagger has posted the second Monday Math Madness contest. It has a fun St. Patrick's Day theme. Check it out! You've got a week to solve this problem and send in your well-explained solution. The Blinkdagger guys are giving out $10 in Amazon gift certificate cash to a randomly-selected winner. April 7th will be the next contest at Wild About Math! I'll be giving away something more fun that cash next time so check back here but solve the Blinkdagger St. Patty's Day problem first. We interrupt your regularly scheduled blog for a non-Math post ... It's Saturday night here, I was just memed by Robert at Reason-4-Smile, I don't feel like doing any productive work so I'll answer the meme and launch it further into the blogosphere. The meme asks me to give a link to the person who meme'd me, to tell 7 interesting things about me, and to then pass the meme on. I interpreted the meme to mean that I should tell things that you're not likely to guess. Here goes: Today is March 14th, or 3/14 as Americans write dates. Think that pi is roughly 3.14 and you'll see why today is Pi Day. Denise at Let's Play Math does a superb job of writing about this special day. She's got a Pi poem, lots of links to pi-related pages and even a hokey but very funny mathematical pi song. Math Mom is also commemorating this special day with some ideas about how to celebrate the day. Plus, there are other ideas in the comments of Math Mom's post. I wanted to state what some of you may have been noticing: My rate of posting has decreased, posts have been more superficial, and I've been slow to respond to comments. Life has been busier than I expected, with important life matters and paid short-term project work filling a significant percent of my time. The slowness may last several weeks. I'm not dropping this blog as I enjoy the value that I give and receive through this community. I may not get to do a significant post until early next week. Stay tuned for the Blinkdagger contest post Monday, and for my next contest the first Monday of April. Also, do subscribe to my mailing list to get Math Bite emails with interesting little Math tidbits. In upcoming posts I'll share a very nice alternative approach to the Monday Math Madness contest I received and I'll review an outstanding Geometry book for those of you who want a real work out when doing Geometry. I'm also going to experiment with doing some short posts that tell you about what other Math bloggers are writing about to keep the momentum moving. And, if any of you would like to write some guest posts to bring some attention to your blogs I'm very open to that possibility. For the very first Monday Math Madness contest we got 13 submissions. Of the 13, 6 were correct. For the record, I solved the problem by enumerating the various cases where 3 categories were represented and computing and adding their probabilities. I also verified my solution to the problem by writing a computer program to enumerate all 243 (3^5) permutations of 3 categories and 5 blog posts and count the ones were all 3 blog categories were represented. So, I'm pretty confident I got the right answer

In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. The surface under consideration may be a closed one enclosing a volume such as a spherical surface. |Part of a series of articles about| The law was first formulated by Joseph-Louis Lagrange in 1773, followed by Carl Friedrich Gauss in 1813, both in the context of the attraction of ellipsoids. It is one of Maxwell's four equations, which form the basis of classical electrodynamics. Gauss's law can be used to derive Coulomb's law, and vice versa. In words, Gauss's law states that Gauss's law has a close mathematical similarity with a number of laws in other areas of physics, such as Gauss's law for magnetism and Gauss's law for gravity. In fact, any inverse-square law can be formulated in a way similar to Gauss's law: for example, Gauss's law itself is essentially equivalent to the inverse-square Coulomb's law, and Gauss's law for gravity is essentially equivalent to the inverse-square Newton's law of gravity. The law can be expressed mathematically using vector calculus in integral form and differential form; both are equivalent since they are related by the divergence theorem, also called Gauss's theorem. Each of these forms in turn can also be expressed two ways: In terms of a relation between the electric field E and the total electric charge, or in terms of the electric displacement field D and the free electric charge. Equation involving the E field where ΦE is the electric flux through a closed surface S enclosing any volume V, Q is the total charge enclosed within V, and ε0 is the electric constant. The electric flux ΦE is defined as a surface integral of the electric field: Since the flux is defined as an integral of the electric field, this expression of Gauss's law is called the integral form. An important fact about this fundamental equation often doesn't find a mention in expositions that are not absolutely diligent. The above equation may fail to hold true in case the closed surface S contains a singularity of the electric field, which is physicists' term for a point in space where either a point charge exists and the field strength approaches infinity, or the field's magnitude or direction gets altered discontinuously due to the existence of a surface charge. In 2011, a modification of the above equation, called the Generalized Gauss's Theorem by its original creator, was published in the proceedings of the 2011 Annual Meeting of Electrostatics Society of America. The Generalized Gauss's Theorem allows the closed surface S to pass through singularities of the electric field. A corollary of the Generalized Gauss's Theorem, known as the simplest form of the Generalized Gauss's Theorem, holds true if the surface S is smooth. It states that where Q is the net charge enclosed within V and Q' is the net charge contained by the closed surface S itself. Applying the integral form If the electric field is known everywhere, Gauss's law makes it possible to find the distribution of electric charge: The charge in any given region can be deduced by integrating the electric field to find the flux. The reverse problem (when the electric charge distribution is known and the electric field must be computed) is much more difficult. The total flux through a given surface gives little information about the electric field, and can go in and out of the surface in arbitrarily complicated patterns. An exception is if there is some symmetry in the problem, which mandates that the electric field passes through the surface in a uniform way. Then, if the total flux is known, the field itself can be deduced at every point. Common examples of symmetries which lend themselves to Gauss's law include: cylindrical symmetry, planar symmetry, and spherical symmetry. See the article Gaussian surface for examples where these symmetries are exploited to compute electric fields. By the divergence theorem, Gauss's law can alternatively be written in the differential form: Equivalence of integral and differential forms The integral and differential forms are mathematically equivalent, by the divergence theorem. Here is the argument more specifically. Outline of proof The integral form of Gauss' law is: for any closed surface S containing charge Q. By the divergence theorem, this equation is equivalent to: for any volume V containing charge Q. By the relation between charge and charge density, this equation is equivalent to: for any volume V. In order for this equation to be simultaneously true for every possible volume V, it is necessary (and sufficient) for the integrands to be equal everywhere. Therefore, this equation is equivalent to: Thus the integral and differential forms are equivalent. Equation involving the D field Free, bound, and total charge The electric charge that arises in the simplest textbook situations would be classified as "free charge"—for example, the charge which is transferred in static electricity, or the charge on a capacitor plate. In contrast, "bound charge" arises only in the context of dielectric (polarizable) materials. (All materials are polarizable to some extent.) When such materials are placed in an external electric field, the electrons remain bound to their respective atoms, but shift a microscopic distance in response to the field, so that they're more on one side of the atom than the other. All these microscopic displacements add up to give a macroscopic net charge distribution, and this constitutes the "bound charge". Although microscopically all charge is fundamentally the same, there are often practical reasons for wanting to treat bound charge differently from free charge. The result is that the more fundamental Gauss's law, in terms of E (above), is sometimes put into the equivalent form below, which is in terms of D and the free charge only. This formulation of Gauss's law states the total charge form: where ΦD is the D-field flux through a surface S which encloses a volume V, and Qfree is the free charge contained in V. The flux ΦD is defined analogously to the flux ΦE of the electric field E through S: The differential form of Gauss's law, involving free charge only, states: where ∇ · D is the divergence of the electric displacement field, and ρfree is the free electric charge density. Equivalence of total and free charge statements Proof that the formulations of Gauss's law in terms of free charge are equivalent to the formulations involving total charge. In this proof, we will show that the equation is equivalent to the equation Note that we are only dealing with the differential forms, not the integral forms, but that is sufficient since the differential and integral forms are equivalent in each case, by the divergence theorem. We introduce the polarization density P, which has the following relation to E and D: and the following relation to the bound charge: Now, consider the three equations: The key insight is that the sum of the first two equations is the third equation. This completes the proof: The first equation is true by definition, and therefore the second equation is true if and only if the third equation is true. So the second and third equations are equivalent, which is what we wanted to prove. Equation for linear materials for the integral form, and for the differential form. In terms of fields of force Gauss's theorem can be interpreted in terms of the lines of force of the field as follows: The flux through a closed surface is dependent upon both the magnitude and direction of the electric field lines penetrating the surface. In general a positive flux is defined by these lines leaving the surface and negative flux by lines entering this surface. This results in positive charges causing a positive flux and negative charges creating a negative flux. These electric field lines will extend to infinite decreasing in strength by a factor of one over the distance from the source of the charge squared. The larger the number of field lines emanating from a charge the larger the magnitude of the charge is, and the closer together the field lines are the greater the magnitude of the electric field. This has the natural result of the electric field becoming weaker as one moves away from a charged particle, but the surface area also increases so that the net electric field exiting this particle will stay the same. In other words the closed integral of the electric field and the dot product of the derivative of the area will equal the net charge enclosed divided by permittivity of free space. Relation to Coulomb's law Deriving Gauss's law from Coulomb's law Strictly speaking, Gauss's law cannot be derived from Coulomb's law alone, since Coulomb's law gives the electric field due to an individual point charge only. However, Gauss's law can be proven from Coulomb's law if it is assumed, in addition, that the electric field obeys the superposition principle. The superposition principle says that the resulting field is the vector sum of fields generated by each particle (or the integral, if the charges are distributed smoothly in space). Outline of proof Coulomb's law states that the electric field due to a stationary point charge is: - er is the radial unit vector, - r is the radius, |r|, - ε0 is the electric constant, - q is the charge of the particle, which is assumed to be located at the origin. Using the expression from Coulomb's law, we get the total field at r by using an integral to sum the field at r due to the infinitesimal charge at each other point s in space, to give where δ(r) is the Dirac delta function, the result is Using the "sifting property" of the Dirac delta function, we arrive at which is the differential form of Gauss' law, as desired. Note that since Coulomb's law only applies to stationary charges, there is no reason to expect Gauss's law to hold for moving charges based on this derivation alone. In fact, Gauss's law does hold for moving charges, and in this respect Gauss's law is more general than Coulomb's law. Proof (without Dirac Delta) Let be a bounded open set, and be the electric field, with a continuous function (density of charge). it's true that . But because , = 0 for the argument above ( and then ) And so the flux through a closed surface generated by some charge density outside (the surface) is null. Let's consider now , and as the sphere centered in having as radius (it exists because is an open set). Let and be the electric field created inside \ outside the sphere: = , = and + = The last equality follows by observing that , and the argument above. The RHS is the electric flux generated by a charged sphere, and so: Where the last equality follows by the mean value theorem for integrals. Finally for the Squeeze theorem and the continuity of : Deriving Coulomb's law from Gauss's law Strictly speaking, Coulomb's law cannot be derived from Gauss's law alone, since Gauss's law does not give any information regarding the curl of E (see Helmholtz decomposition and Faraday's law). However, Coulomb's law can be proven from Gauss's law if it is assumed, in addition, that the electric field from a point charge is spherically symmetric (this assumption, like Coulomb's law itself, is exactly true if the charge is stationary, and approximately true if the charge is in motion). Outline of proof Taking S in the integral form of Gauss' law to be a spherical surface of radius r, centered at the point charge Q, we have By the assumption of spherical symmetry, the integrand is a constant which can be taken out of the integral. The result is where r̂ is a unit vector pointing radially away from the charge. Again by spherical symmetry, E points in the radial direction, and so we get which is essentially equivalent to Coulomb's law. Thus the inverse-square law dependence of the electric field in Coulomb's law follows from Gauss' law. - Duhem, Pierre. Leçons sur l'électricité et le magnétisme (in French). vol. 1, ch. 4, p. 22–23. shows that Lagrange has priority over Gauss. Others after Gauss discovered "Gauss' Law", too. - Lagrange, Joseph-Louis (1773). "Sur l'attraction des sphéroïdes elliptiques". Mémoires de l'Académie de Berlin (in French): 125. - Gauss, Carl Friedrich. Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata (in Latin). (Gauss, Werke, vol. V, p. 1). Gauss mentions Newton's Principia proposition XCI regarding finding the force exerted by a sphere on a point anywhere along an axis passing through the sphere. - Halliday, David; Resnick, Robert (1970). Fundamentals of Physics. John Wiley & Sons. pp. 452–453. - Serway, Raymond A. (1996). Physics for Scientists and Engineers with Modern Physics (4th ed.). p. 687. - Grant, I. S.; Phillips, W. R. (2008). Electromagnetism. Manchester Physics (2nd ed.). John Wiley & Sons. ISBN 978-0-471-92712-9. - Matthews, Paul (1998). Vector Calculus. Springer. ISBN 3-540-76180-2. - Pathak, Ishnath (2011). "A Generalization of Gauss's Theorem in Electrostatics". Proceedings of the 2011 ESA Annual Meeting on Electrostatics: C3. - See, for example, Griffiths, David J. (2013). Introduction to Electrodynamics (4th ed.). Prentice Hall. p. 50. - MIT Video Lecture Series (30 x 50 minute lectures)- Electricity and Magnetism Taught by Professor Walter Lewin. - section on Gauss's law in an online textbook - MISN-0-132 Gauss's Law for Spherical Symmetry (PDF file) by Peter Signell for Project PHYSNET. - MISN-0-133 Gauss's Law Applied to Cylindrical and Planar Charge Distributions (PDF file) by Peter Signell for Project PHYSNET.

A triangle has no one unique center, but the circumcenter may be the second most popular and easy to visualize, after the incenter. The circumcircle is the smallest circle that can fit through the three points that define a triangle. The circumcircle has a radius, R, that is equal to a*b*c/(4K), where K is the area of the triangle, and a, b, and c are the side lengths of the triangle ΔABC. We will denote the circumcenter as O The circumcenter's coordinates are: (dA*(Cy-By) + dB*(Ay-Cy) + dC*(By-Ay)) Ox = ----------------------------------------- (1a) 2*(Ax*(Cy-By) + Bx*(Ay-Cy) + Cx*(By-Ay)) -(dA*(Cx-Bx) + dB*(Ax-Cx) + dC*(Bx-Ax)) Oy = ----------------------------------------- (1b) 2*(Ax*(Cy-By) + Bx*(Ay-Cy) + Cx*(By-Ay)) O = (Ox, Oy)... Cartesian coordinates of the circumcenter A = (Ax, Ay)... the coordinates of vertex A of triangle ABC B = (Bx, By)... the coordinates of vertex B of triangle ABC C = (Cx, Cy)... the coordinates of vertex C of triangle ABC and where some intermediate calculations help reduce eyestrain: dA = Ax^2 + Ay^2 (2a) dB = Bx^2 + By^2 (2b) dC = Cx^2 + Cy^2 (2c) The best explanation for finding the center of the circle is found here. The Khan Academy always has marvelous tutorials on YouTube, and they also explain this quite well here.. Example 1: An acute triangle has vertices A, B, and C at A = (-2,-2), B = (5,3), and C = (1,4). Find the circumcenter O and the radius of the circumcircle, R. A = (Ax, Ay) = (-2,-2) B = (Bx, By) = ( 5, 3) C = (Cx, Cy) = ( 1, 4) dA = (-2)^2 + (-2)^2 = 8 dB = 5^2 + 3^2 = 34 dC = 1^2 + 4^2 = 17 O = (Ox, Oy) = (2.06, -0.28) R = 4.4 The result is that the circumcenter is found at (2.06, -0.28) and the radius of the circumcircle is 4.4. Point of Concurrency of Perpendicular Bisectors: The circumcenter is the point of concurrency of the perpendicular bisectors of each side. If you bisect every side, and you draw the line that runs perpendicular to that side then every line intersects at one point: the circumcenter. When I began this writeup, I was under the impression that Euclid had proved that the perpendicular bisectors from every side all meet at the same point for every possible triangle. He showed something similar in Book 4, Proposition 5 of The Elements. But Cut the Knot mathematician and author Alexander Bogomolny says that Euclid didn't do this. He only showed that they did, but offered no proof. It's Bogomolny's belief that Euclid would have needed a Ceva's Theorem in order to prove it, but that theorem didn't come along for another 1500 years. SOURCE: Jim Wilson, Proof that the three perpendicular biectors of the sides of a triangle are concurrent. Wilson is a professor with the Mathematics Education program at the University of Georgia. His web site is full of mathematical topics. The Circumcenter is Outside the Triangle for Obtuse Triangles: Although the incenter is always inside the triangle, the circumcenter does not have to be. When the triangle is acute, the circumcenter is inside ΔABC. When it is obtuse, O is outside. Example 2 gives points of a very small obtuse angle with a wide vertex angle at A. The circumcenter is a large distance away from the triangle. Example 2: An obtuse triangle has one vertex at the origin. We'll label this vertex A. The triangle is isosceles, meaning that sides AB and AC are of equal length. The interior angle α is 150°. The vertex coordinates are: A = (0,0), B = (1,0), and C = (cos(150°),sin(150°)) = (-0.87, 0.5). Find the circumcenter and the radius of the circumcircle. A = (Ax, Ay) = ( 0, 0) B = (Bx, By) = ( 1, 0) C = (Cx, Cy) = (-0.87, 0.5) dA = 3.73 dB = 1 dC = 1 O = (Ox, Oy) = (0.5, 1.87) R = 1.93 The circumcenter is located at O = (0.5, 1.87). The radius is R = 1.93, a comparatively large distance away from the triangle. This is for an interior angle α = 150°. If the interior angle were greater, the radius would be even larger. For α = 170°, R = 5.7. A straightforward equation for the circumcenter was difficult to find on the internet, and when I sat down to derive it myself, I was dismayed at how messy the terms got. When I did find a workable equation (Equation 1 above), I wanted to see if it would work for a variety of cases, and so I dropped the equation and many of its preceding calculational terms into Excel, created a graph of a triangle, the circumcenter point, and then the circumcircle to see if the equation would work and was well behaved and so forth. A picture of Example 1 is on my homenode, and will stay there for a brief time. Barycentric Coordinates: The barycentric coordinates of the circumcenter are sin(2α):sin(2β):sin(2γ). (The interior angles at triangle vertices A, B, and C are α, β, γ, respectively.) Trilinear Coordinates: The trilinear coordinates of the circumcenter are cos(α):cos(β):cos(γ). Everything2 Writeups: Articles on (topic) - tongpoo, circumcenter, Dec. 2, 2001 - tongpoo, circumcircle. A nodeshell was created, but it was never filled. Clearly this hole must be filled! - tongpoo, triangle, Feb. 8, 2002 - IWhoSawTheFace, incenter, Feb. 8, 2002 References: Useful books and references on geometry H.S.M. Coxeter, Introduction to Geometry, 2nd Ed., (c) 1969 *SIGH* What a magnificent book. § 1.4, “The Medians and the Centroid,” p. 10 § 1.5, “The Incircle and the Circumcircle,” pp. 11-16 § 1.6, “The Euler Line and the Orthocenter,” p. 17 Dan Pedoe, Geometry: A Comprehensive Course J.L. Heilbron, Geometry Civilized, ©2000 David Wells, Ed., The Penguin Dictionary of Curious and Interesting Geometry, ©1991 Bruce Meserve, Fundamental Concepts of Geometry, ©1983 Melvin Hausner, A Vector Space Approach to Geometry, ©1965 Gerald Farin and Dianne Hansford, The Geometry Toolbox, ©1998 Ch. 3, 2D Lines § 3.6, “Distance of a point to a line,” p. 40 § 3.7, “The foot of a point,” p. 44 § 3.8, “Computing intersections,” p. 45 Ch. 8, Breaking it up: Triangles § 8.1, “Barycentric coordinates,” p. 126 § 8.2, “Affine invariance,” p. 128 § 8.3, “Some special points,” p. 128 Daniel Zwillinger, Ed., The CRC Standard Mathematical Tables and Formulae, 30th Ed, ©1996 Ch. 4, Geometry, § 4.5.1, “Triangles,” p. 271 § 4.6, “Circles,” p. 277 Siobhan Roberts, King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry, ©2006 Alfred S. Posamentier and Charles T. Salkind, Challenging Problems in Geometry, ©1988 Hans Rademacher and Otto Toeplitz, The Enjoyment of Mathematics, Published in 1957 by the Princeton University Press § 26, “A characteristic property of the circle,” p. 160 § 28, “The indispensability of the compass for the constructions of elementary geometry,” p. 177 - Wikipedia, "Circumscribed Circle" This contains very useful mathematical formulae, especially matrix forms for finding the center of the circle, and exterior angles at the intersections of the circumcircle with the vertices of a triangle. - Wikipedia, "Triangle" - Wikipedia, "Incircle and Excircles of a Triangle" - D. Joyce, Euclid's Elements, Book 4, Proposition 5, "To circumscribe a circle about a given triangle." David Joyce is a professor of Mathematics and Computer Science at Clark University. He rendered Euclid's Elements into HTML, added Java applets to illustrate geometric constructions with live, movable points and lines. If you're a geometry buff, you should bookmark this site. - Jim Wilson, Proof that the three perpendicular biectors of the sides of a triangle are concurrent. Wilson is a professor with the Mathematics Education program at the University of Georgia. His web site is full of mathematical topics. - To construct a circle given three points. Nice Java applet allows you to drag vertices around and watch the circumcenter move. - Weisstein, Eric W. "Circumcenter" From MathWorld--A Wolfram Web Resource. - Weisstein, Eric W. "Circumcircle" From MathWorld--A Wolfram Web Resource. - Weisstein, Eric W. "Circumradius" From MathWorld--A Wolfram Web Resource. - Weisstein, Eric W. "Incenter" From MathWorld--A Wolfram Web Resource. - Weisstein, Eric W. "Triangle" From MathWorld--A Wolfram Web Resource. - Weisstein, Eric W. "Tangential Triangle" From MathWorld--A Wolfram Web Resource. - Alexander Bogomolny, "Incircle and Excircles of a Triangle" From Cut The Knot--mathematical topics. Cut the Knot has a full range of geometric topics.

Heisenberg Uncertainty Principle: True or False? From Bill Howell: I’m interested in how your worldview can potentially be used to empirically test the natural world. As an example, it seems to me that your theory predicts that (someday) the Heisenberg Uncertainty Principle could be proved false. If one interprets that Principle to be about the inability to measure something because any method of measurement will disturb the object being studied, then theoretically, it would be possible to find a way to make a measurement that doesn’t disturb the object. Using the example of waves, one can’t discern the shape of an object with an instrument that uses a larger wavelength than the thing being studied. If particles don’t exist beyond a certain size (let’s say the Plank limit) then there is no way to discriminate the state of a subatomic particle that can be influenced by an interaction on that scale of magnitude. But if particles can exist which are infinitely small, then (theoretically) there are particles that are a magnitude or two less than the Plank limit which could be used to probe the structure of things at and above the Plank limit. [Bill, I hate to disappoint you, but the Heisenberg Uncertainty Principle cannot be proven false. At best, one can choose one of the two possible interpretations that can be gained from it: 1.) Uncertainty means that nature contains an element of absolute chance (Copenhagen view); 2.) Uncertainty reflects observer ignorance (Bohmian view). With UNCERTAINTY (It is impossible to know everything about anything, but it is possible to know more about anything), we have chosen the second along with its implied challenge to the finite causality of classical mechanism. This does not mean, however, that one can perform any measurement without disturbing the microcosm being measured. All microcosms contain an infinity of submicrocosms and are bathed in an infinity of supermicrocosms, so, of course, the Plank limit is only defined by those used to perform the measurement. You are right that the use of still smaller microcosms (ether particles?) would allow measurements at even smaller scales. Incidentally, the “wave nature” of particles being measured at the scale to which the principle is applied is due to the motions induced within the macrocosm of any such particle. Interpretations differ because positivists, in particular, deny that the associated macrocosm contains anything at all. For them, the surrounding space is perfectly empty, and so they see related waves as properties of the microcosm itself. In our view, however, any particle traveling through the macrocosm must produce waves in the same way that a ship makes waves as it travels across the ocean. There is no “wave-particle” duality in univironmental determinism (UD).] The fact that we currently have no instruments capable of probing at this scale is beside the point. One can’t say it’s impossible to create such an instrument someday in the future (because we are still too ignorant about the Cosmos to say that it’s impossible). Creating such an instrument may require using subatomic particles that are very close to absolute zero. Conversely, if one interprets the Heisenberg Uncertainty Principle to mean that it is not about the inability to physically measure something (which is an alternate explanation I’ve read), then that too is opposed to UD (as I understand it) and so such an instrument, if it could be created, would also falsify the Principle. [You are right that “the inability to physically measure something” and “the inability to know everything about anything” really are equivalent. Nonetheless, a better instrument would only falsify the Planck limit, but not the principle. According to our assumption of INFINITY, the principle will apply each time a new “quasi-Planck limit” is reached.] On a separate matter, you once told me that you weren’t a ‘steady-state’er. I had knee-jerked assumed that this was your conceptual model of the Universe. So I’m curious, is your position based on what might be called ‘first principles’ and/or the logic of your 10 Assumptions, or do you have a conceptual model that you could describe to me. [Remember that the Steady State Theory (SST) proposed first by Bondi and Gold (1948) involved the assumption of creation, which is the opposite of our deterministic assumption of CONSERVATION (Matter and the motion of matter can be neither created nor destroyed). They did this to stay in tune with the prevailing view that the universe was expanding. The creation of one hydrogen atom per cubic meter per billion years was calculated to be enough to keep the universe expanding forever. Bondi and Gold did not mention what the universe supposedly was expanding into. Neither did they use the 4-D concept of space-time that Einstein had introduced. As you know, UD assumes through INFINITY (The universe is infinite, both in the microcosmic and macrocosmic directions) that empty space cannot exist. It also assumes through INSEPARABILITY (Just as there is no motion without matter, so there is no matter without motion) that there are only three dimensions, and that the objectification of time in SRT and GRT is Einstein’s greatest philosophical error (http://thescientificworldview.blogspot.com/2010/10/einsteins-most-important-philosophical.html). A short paper on my conceptual model of Infinite Universe Theory (IUT) is at: http://scientificphilosophy.com/Downloads/IUT.pdf . I am expanding this into a short, easy to understand book for the layperson. Bondi, H., and Gold, T., 1948, The steady-state theory of the expanding universe: Monthly Notices of the Royal Astronomical Society, v. 108, no. 3, p. 252-270.

Key Concepts And Summary The behavior of gases can be described by several laws based on experimental observations of their properties. The pressure of a given amount of gas is directly proportional to its absolute temperature, provided that the volume does not change . The volume of a given gas sample is directly proportional to its absolute temperature at constant pressure . The volume of a given amount of gas is inversely proportional to its pressure when temperature is held constant . Under the same conditions of temperature and pressure, equal volumes of all gases contain the same number of molecules . The equations describing these laws are special cases of the ideal gas law, PV = nRT, where P is the pressure of the gas, V is its volume, n is the number of moles of the gas, T is its kelvin temperature, and R is the ideal gas constant. Bodies In Thermodynamic Equilibrium For experimental physics, hotness means that, when comparing any two given bodies in their respective separate thermodynamic equilibria, any two suitably given empirical thermometers with numerical scale readings will agree as to which is the hotter of the two given bodies, or that they have the same temperature. This does not require the two thermometers to have a linear relation between their numerical scale readings, but it does require that the relation between their numerical readings shall be strictly monotonic. A definite sense of greater hotness can be had, independently of calorimetry, of thermodynamics, and of properties of particular materials, from Wien’s displacement law of thermal radiation: the temperature of a bath of thermal radiation is proportional, by a universal constant, to the frequency of the maximum of its frequency spectrum this frequency is always positive, but can have values that tend to zero. Thermal radiation is initially defined for a cavity in thermodynamic equilibrium. These physical facts justify a mathematical statement that hotness exists on an ordered one-dimensional manifold. This is a fundamental character of temperature and thermometers for bodies in their own thermodynamic equilibrium. Conditions For Standard Temperature And Pressure The standard temperature and pressure are the conditions used in thermodynamics to figure out the reference points of various gases. These are also used to specify the vapor volume. The gas volumes can be converted to the number of moles which makes it easier to find the reference points. The volume of a gas varies with temperature and pressure hence it makes it impossible to measure the molar quantity. When the gas volume is fixed to its temperature and pressure, then the standard volume is directly proportional to the molar quantity. With the help of standard temperature and pressure, the compositions of gases can also be calculated. The value of temperature and pressure depends on the organization which defines them. The standard pressure is equal to the atmospheric pressure and the temperature is close to room temperature. NTP is a term that is referred to as normal temperature and pressure. The NTP uses 20°C as the standard temperature whereas the standard temperature and pressure use 0°C as the standard temperature. The NTP was used by the National Institute of Standards and Technology. The calculation of volume at NTP is given as You May Like: What Is The Definition Of Mantle In Geography Breathing And Boyles Law What do you do about 20 times per minute for your whole life, without break, and often without even being aware of it? The answer, of course, is respiration, or breathing. How does it work? It turns out that the gas laws apply here. Your lungs take in gas that your body needs and get rid of waste gas . Lungs are made of spongy, stretchy tissue that expands and contracts while you breathe. When you inhale, your diaphragm and intercostal muscles contract, expanding your chest cavity and making your lung volume larger. The increase in volume leads to a decrease in pressure . This causes air to flow into the lungs . When you exhale, the process reverses: Your diaphragm and rib muscles relax, your chest cavity contracts, and your lung volume decreases, causing the pressure to increase , and air flows out of the lungs . You then breathe in and out again, and again, repeating this Boyles law cycle for the rest of your life . Bodies In A Steady State But Not In Thermodynamic Equilibrium While for bodies in their own thermodynamic equilibrium states, the notion of temperature requires that all empirical thermometers must agree as to which of two bodies is the hotter or that they are at the same temperature, this requirement is not safe for bodies that are in steady states though not in thermodynamic equilibrium. It can then well be that different empirical thermometers disagree about which is hotter, and if this is so, then at least one of the bodies does not have a well-defined absolute thermodynamic temperature. Nevertheless, anyone has given body and any one suitable empirical thermometer can still support notions of empirical, non-absolute, hotness, and temperature, for a suitable range of processes. This is a matter for study in non-equilibrium thermodynamics. Don’t Miss: Algebra 2 Sequences And Series What Is Standard Temperature And Pressure Some textbooks define it differently than others, but the newest IUPAC standard temperature and pressure are: Some older textbooks might say: The difference between the two pressures are subtle but significant: #”1 bar” = 1.00000xx10^5# This leads to a difference of about #”0.3 L”# of an ideal gas at STP when you calculate it using the Ideal Gas Law: #~”22.7 L”# #~”22.4 L”# for the latter. Ever had your university lab notebook “torn apart” by a lab TA for “not enough information”? Yeah, it’s primarily because science tends to rely on consistency and reproducibility to prove that something is credible. If someone can’t read your lab notebook and then reproduce your lab experiment without your input and correction, you haven’t provided enough information to replicate that experiment precisely. IUPAC has defined such standards so that people have consistent atmospheric conditions to use for comparisons of data from different experimental trials for the same type of experiment. That improves the accuracy to which an experiment can be reproduced. SIDENOTE: This is not be confused with the temperature and pressure at which #DeltaH_f^@# Overview Of Standard Temperature And Pressure STP is an abbreviation for standard temperature and pressure. It is a set of conditions for temperature and pressure measurements for physical and chemical reactions. The standard temperature is zero º Celsius, which corresponds to 32 °F or 273.15 K. The standard pressure is 1 bar or 100.000 kPa. The molar volume of a gas is important for identifying th conditions of temperature and pressure when stating the molar volume of a gas. The molar volume of gas can be calculated by the universal gas law. The temperature and pressure are state functions. Temperature changes according to the location and season, whereas pressure depends on the weather conditions and the height above the sea level. STP is not a standard state condition. A standard state is used in thermodynamics calculations. You May Like: What Is The Electron Geometry Of So3 Definitions Of Standard Temperature And Pressure The term STP means different things to different people and can cause problems in the presentation of adsorption data, because the most common units for the ordinates of such plots are standard volumes per unit mass of adsorbent. The International Union of Pure and Applied Chemistry used to define STP as 0 C and 1 atm . This definition is now obsolete. The preferred definition, since 1982, is 273.15 K and 1 bar . The National Institute of Standards and Technology, on the other hand, defines STP as 1 atm and 20 C . In practical applications of adsorption equipment, STP often refers to the pressure of the room and the temperature of the adsorption manifold, which is often kept above room temperature to prevent changes in room temperature from changing the apparent manifold or dead volumes . Room temperature, at least in the summer months in many places, is usually close to 25 C, which is a de facto standard in gas flow controllers. For the sake of comparison, we recommend specifying the definitions of STP being used if there is any concern that they will affect the results. The difference between using a standard temperature of 0 C and 25 C will, in the authors’ experience, produce errors in the volume adsorbed that are less than the error in the measurements themselves. In commercial equipment, the volume adsorbed is usually reported by defining STP as 0 C and 760 Torr . Alain Tressaud, in, 2019 What Are The Conditions For Stp In Chemistry In respect to this, is STP 25 or 0? Both STP and standard state conditions are commonly used for scientific calculations. STP stands for Standard Temperature and Pressure. It is defined to be 273 K and 1 atm pressure . Temperature is not specified, although most tables compile data at 25 degrees C . Subsequently, question is, what does STP stand for? Standard Temperature and Pressure In this way, what is the standard value for pressure at STP condition? Standard Temperature and Pressure. Standard temperature is equal to 0 °C, which is 273.15 K. Standard Pressure is 1 Atm, 101.3kPa or 760 mmHg or torr. STP is the standardconditions often used for measuring gas density and volume. What is PV nRT called? PV = nRT: The Ideal Gas Law. Fifteen ExamplesEach unit occurs three times and the cube root yields L-atm / mol-K, the correct units for R when used in a gas law context. Consequently, we have: PV / nT = R. or, more commonly: PV = nRT. R is the gas constant. Also Check: Chapter 9 Review Algebra 2 What Is Standard Temperature And Pressure In Chemistry - Standard temperature and pressure, abbreviated STP, refers to nominal conditions in the atmosphere at sea level. This value is important to physicists, chemists, engineers, and pilots and navigators. - Standard temperature is defined as zero degrees Celsius , which translates to 32 degrees Fahrenheit or 273.15 degrees kelvin . This is essentially the freezing point of pure water at sea level, in air at standard pressure. - Standard pressure supports 760 millimeters in a mercurial barometer . This is about 29.9 inches of mercury, and represents approximately 14.7 pounds per inch . Imagine a column of air measuring one inch square, extending straight up into space beyond the atmosphere. The air in such a column would weigh about 14.7 pounds. - The density of air at STP is approximately 1.29 kilogram per meter cubed . This fact comes as a surprise to many people a cubic meter of air weighs nearly three pounds! - See also: SI , specific gravity. Chemistry End Of Chapter Exercises the appropriate graph Read Also: Electron Pair Geometry Of H2o Why Do We Need Satp SATP is brought by the International Union of Pure and Applied Chemistry . It is used in Chemistry as a reference standard condition. The standard ambient temperature in SATP is 25 . This temperature is more practical and convenient compared to 0 of STP . It would be easier for chemists all over the world to take SATP conditions as a reference point compared to STP conditions. Why Do We Need Standards Chemists require STP definitions because the behavior of a substance varies greatly depending on the temperature and pressure. STP definitions give chemists a common reference point to describe how a gas behaves under normal conditions. Scientists use standards like STP definitions for two purposes, to define certain quantitative metrics and to allow for consistent and repeatable experiments. Imagine someone tells you the molar volume of methane is 22.4 liters . The molar volume of a substance is just a measure of how much space one mole of that substance takes up. On its own, this value is not very informative. It is known that the volume of a gas varies greatly with respect to pressure and temperature, so gas could have multiple molar volumes, depending on the exact temperature and pressure. One needs to specify a temperature and pressure to make a molar volume measurement of 22.4 L a more meaningful quantity. Scientists agree upon a predefined temperature and pressure to report quantitative properties of gases. As it just so happens, one mole of any gas at STP has a volume of 22.4 L. Quantitative measurements of gas, like volume, volumetric flow, and compressibility, all must be defined with respect to some defined pressure and temperature. The true method of knowledge is experiment. William Blake Also Check: What Does Cohesion Mean In Biology Correct Use Of The Term Stp Even though STP is defined, you can see the precise definition depends on the committee that set the standard! Therefore, rather than citing a measurement as performed at STP or standard conditions, it’s always best to explicitly state the temperature and pressure reference conditions. This avoids confusion. In addition, it is important to state the temperature and pressure for the molar volume of a gas, rather than citing STP as the conditions. When calculated molar volume, one should state whether the calculation used the ideal gas constant R or the specific gas constant Rs. The two constants are related where Rs = R / m, where m is the molecular mass of a gas. Although STP is most commonly applied to gases, many scientists try to perform experiments at STP to SATP to make it easier to replicate them without introducing variables. It’s good lab practice to always state the temperature and pressure or to at least record them in case they turn out to be important. Volume And Pressure: Boyles Law If we partially fill an airtight syringe with air, the syringe contains a specific amount of air at constant temperature, say 25 °C. If we slowly push in the plunger while keeping temperature constant, the gas in the syringe is compressed into a smaller volume and its pressure increases if we pull out the plunger, the volume increases and the pressure decreases. This example of the effect of volume on the pressure of a given amount of a confined gas is true in general. Decreasing the volume of a contained gas will increase its pressure, and increasing its volume will decrease its pressure. In fact, if the volume increases by a certain factor, the pressure decreases by the same factor, and vice versa. Volume-pressure data for an air sample at room temperature are graphed in Figure 5. Unlike the PT and VT relationships, pressure and volume are not directly proportional to each other. Instead, P and V exhibit inverse proportionality: Increasing the pressure results in a decrease of the volume of the gas. Mathematically this can be written: The relationship between the volume and pressure of a given amount of gas at constant temperature was first published by the English natural philosopher Robert Boyle over 300 years ago. It is summarized in the statement now known as Boyles law: The volume of a given amount of gas held at constant temperature is inversely proportional to the pressure under which it is measured. Don’t Miss: What Is C5 In Chemistry Physical Chemical Characteristics Of Fluorine Table 1. Physicalchemical data of fluorine On March 23, 1962, when platinum hexafluoride, a red gas, was allowed to mix with a large molar excess of xenon , the immediately formed product was a yellow solid with the composition Xe+ , the first synthesized compound of a noble gas.99 Fig. 27. Synthesis of the first compound of noble gases by N. Bartlett: Xe and PtF6 gases prior to reaction resulting XePtF6. This discovery was acclaimed101 as one of the ten most beautiful experiments in chemistry, because102 it opened the way for the noble gas chemistry and high oxidation species chemistry, as illustrated by the various species isolated in Neil’s Berkeley Group: first example of Os + VII: OsOF5 NOWF7, 2WF8, NOReF7 Isolation of ONF3: 2NiF6 ONF + ONF3 + NiF2 synthesis and oxidizing properties of ReOF5 and OsOF5 salts of quinquevalent gold: M+ AuF6, NO+ AuF6, Xe2F11+ AuF6, XeF5+ AuF6, etc. Quoting P. Ball, we can say that Neil Bartlett’s experiment tells us that in chemistry, the wonders never cease.103 Now, fluorine gas is produced following the electrochemical process, as shown, for instance at Fig. 28 for the F2 synthesis facilities of Advance Research Chemicals, Inc. . An important technical point is that hooded areas should be specially equipped for fluoride manufacturing. All work should be done in hoods, 40,000 cfm fresh air being pulled in to avoid depression in the building. Fig. 28. Fluorination facilities at ARC, Inc.

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Calculus covers all topics from a typical high school or first-year college calculus course, including: Problems on the limit of a function as x approaches a fixed constant ; limit of a function as x approaches plus or minus infinity My math teacher teaches all levels of math from to calculus and he teaches me now so im ready because i asked him for help but its still hard to understand calculus and what the equation equals to. Get an introduction to the essential mathematical knowledge and skills required to take a first course in calculus. Another private goal in this area is the study of ambimorphic symmetries. With the ability to answer questions from single and multivariable calculus, Wolfram Alpha is a great tool for computing limits, derivatives and integrals and their applications, including tangent lines, extrema, arc length and much more. With more than 2. The Calculus I varbery of many of the problems tends to be skipped and left to the student to verify or fill in the details. This book is an outgrowth of our teaching of calculus at Berkeley, and the present edition incorporates many improvements based on our use of the first edition. Calculus II will cover sections 5, 6, 7, and 8 of the textbook, starting with section 5. This calculus course covers differentiation and integration calculhs functions of one variable, and concludes with a brief discussion of infinite series. Calculus help number 8 in for colleges The early help calculus tradition of wisdom on a slippery slope if looking to reshape everyday behaviors associated with the process which moves it can get started. I may keep working on this document as the course goes on, so these notes will not be completely Section 9. Calculus Exams From Previous Semesters. Krista King is an Calcylus Calculus 1 Lecture Videos These lecture videos are organized in an order that varbegg with the current book we are using for our Math, Calculus 1, courses Calculus, with Differential Varnerg, by Varberg, Purcell and Rigdon, 9th edition published slution Pearson. Guichard Calculus help number 8 in for colleges The early help calculus tradition of wisdom on a slippery slope if looking to reshape everyday behaviors associated with the process which moves it can get started. The surprise is that something seemingly so abstract ends up explaining the real world. Business math lecture However when it comes to Calc II, you might look at a problem and find a much easier solution. These mathematical tools and methods are used extensively in the physical sciences, engineering, economics and computer rigron. Problems on the limit of a function as x approaches a fixed constant ; limit of a function as x approaches plus or minus infinity Single and Multivariable Calculus Lecture Notes by D. Unlike the Calculus AB examit also covers parametric, polar, and vector functions. Exam 1 without solutions. Math Calculus 3 Lecture Videos These lecture videos are organized in an order that corresponds with the current book we are using for our Math, Calculus 3, courses Calculus, with Differential Equations, by Varberg, Purcell and Rigdon, 9th edition published by Pearson. The pdf files for this current semester are posted at the UW calculus student page. Learn calculus the easy way – Purchase a DVD set of easy to follow instructions – You control the speed of learning. Video Tutorials are downloadable to watch Offline Precalculus review and Calculus preview – Shows Precalculus math in the exact way you’ll use it for Calculus – Also gives a preview to many Calculus concepts. It is the second ridon in the freshman calculus sequence. This class counts for a total of 12 credits. Calculus from Latin calculus, literally ‘small pebble’, used for counting and calculations, as on an abacus is rigdom mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. Marriage of Differential and Integral Calculus. The first reason is that this course does require pucrell you have a very good working knowledge of Calculus I. Calculus with Analytic Geometry by – Dartmouth College Precalculus review and Calculus preview – Shows Precalculus math in the exact way you’ll use it for Calculus – Also gives a preview to many Calculus concepts. Save on the cost of your degree. But, I passed it taking it as a summer course, which means we covered just as much material, but in a much shorter period of time – vraberg summer semester. The MIT course was designed for MIT students who have mostly already learned calculus on some level, so the lectures aren’t necessarily easy to follow if you’re just starting out. The mission of MIT is solutkon advance knowledge and educate students in science, technology, and other areaThe gradient is a fancy word for derivative, or the rate of change of a function. Varbeerg note that doing the lessons listed below will ccalculus count towards your grade. You may find it helpful to consult other texts or information on the internet for additional information. Show your work and justify your answer. The videos listed here are an example of some of the useful videos on KhanAcademy. There are lots of exercises and examples. Process calculi provide a tool for the high-level description of interactions, communications, and synchronizations between a collection of independent agents or processes. 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Shadow of helix on coordinate planes, curves as intersection of two surfaces, curves embedded on a surface, introduction to computer aided graphing via Mathematica 6. Calculus I, or equivalent. The sample tests are just to give you an idea of the a general idea of the topics covered, the level of difficulty, how questions may be worded and, if solutions are provided, what is the acceptable level of detail required in the solutions.

Quickly convert miles/hour into kilometers/hour (MPH to km/h) using the online calculator for metric conversions and more.The answer is 0.62137119223733. We assume you are converting between mile /hour and kilometre/hour. 130 km/h 80 mph. Speed limit on motorways (highways) in most countries of Europe.mph vs. kph (km/h) Example If your speed or velocity is given in miles per hour (mph) and you want to translate it into kilometers per hour (km/h) type your mph value into edit box labeled miles per hour. Conversion Mile per hour to Kilometer per hour (mph to kph). Miles per hour 50 mph 80.467 kph, 80 mph 128.747 kph, 110 mphThis selection will show you how to convert various numbers from the U.S. system of measurement to the miles per hour (mph) into kilometers/hour (km/hr). After that, check the tire pressure and pump more air into the tire if necessary (see Step 2). Then you can proceed to drive with caution for a maximum distance of 200 km (120 miles) and at a maximum speed of 80 km/h (50 mph). Kilometer per hour (km/h) Kilometer per minute (km/min) Mile per hour (mi/ h) Mile per minute (mi/min) Nautical mile per hourWest leaves the crime scene on a motor cycle that goes 80 MPH heading east.XY10, without going into decimals that would leave you with 6 possible answers. The symbol for kilometers per hour is km/h and the International spelling for this unit is kilometres per hour. There are 1.609344 km/h in mph.50 mph. 80.4672 km/h. Air causes a significant resistance at speeds above 80 km/h (50 mph) and is negligible under 30 km/h (20 mph).Exit speed from mound (km/h (mph)) Entry speed into mound (km/h (mph)) Factor given in Table 17.4. Transform 80 kilometers per hour into miles per hour and calculate how many miles per hour is 80 kilometers per hour.Instantly Convert Miles Per Hour (mph) to Kilometres Per Hour (km/h) and Many More Speed Or Velocity Conversions Online. 80 km/h 49.71 mph. Comment from/about : speedy move | Permalink.my average walking speed : 60 meters per minute ( m/min ), converts into : 3.6 kilometers per hour ( km/h ) 3600 meters per hour exactly. 305 km/h / 190 mph GPS-speed G-Power BMW M3 SKII on German Autobahn.80mph?What happens if you let your camera go fast, faster or what happens with 190 mph - 300km/h on the Autobahn with a Corvette C6 ? The chart can be divided into four areas, corresponding to four velocity ranges80 120 km/h (49.71 - 74.56 mph): Within this range as the average speed increases the fuel consumption augments too. 80 KMH is equal to how many MPHHow to recalculate 80 Km per hour to Miles per hour?It would be pertinent to mention here that miles per house perhaps came into existence and Convert between the units (mph km/h) or see the conversion table.50 Miles per hour 80.4672 Kilometres per hour. For my friends from the United States to see the speed on the German Autobahn ) . 200 Km H Into Mph. This is a measurement of speed typically used in countries using the metric system for transport. Road speed limits are given in kilometers per hour which is abbreviated as kph or km/h. A US passenger train that derailed, killing three people, was travelling at 80mph (130km/h) on a curve with a speed limit of 30mph, data from the trains rear engine indicates. It happened in Washington state during rush hour on Monday and officials say 72 people were taken to hospital. Convert 80 Miles/Hour to Kilometers/Hour (mph to km/h) with our conversion calculator and conversion tables.80 mph 128.72570194384 km/h. You also can convert 80 Miles/Hour to other Speed (popular) units. To convert between Mile/hour (mph) and Kilometer/hour (kmh) you have to do the following50 mile/hour (mph) in kilometer/hour (kmh) 80.4672 mph. A Swedish man has been pronounced dead after he and a friend rode a shopping trolley into oncoming traffic in a suburb of Sydney. Police have reported that the trolley reached estimated speeds of up to 50 mph (80km/h) as it was ridden down a steep hill, before colliding with a car at around They are indicated in kilometers per hour. Highway : 130 km/h, if raining 110 km/h (80 mph, 70 mph) Dual cariageway : 110 km/h, ifa town, the reduction of the speed limit from 90 km/h to 50 km/h may not always be indicated by such a road sign, but just by the road sign indicating entry into town. Speed conversions between miles per hour (mph, miph, m/h, mi/h) and kilometres per hour (kph, kmph, km/h) are usually used for calculating speed limits when driving abroad, espcially for UK and US drivers.80.47. mph. kph. Virgo: elongated, pearly tail, shimmering in multiple colours, fanning out into a slightly translucent, silky fluke. Moves through the water slowly and gracefully.The fastest hunter reaching speeds up to 80 km/h or 50 mph. Quickly convert miles into kilometres (70 miles to km) .In North American slang and military usage, km/h . Full mph to kph conversion tables for speeds of .Conversion table for knots to miles per hour Conversion table for . 70: Knots 80.6: MPH: 75: Knots 86.4: MPH: 80: Knots 92.2 In the postwar years, a thicker asphaltic concrete cross-section with full paved hard shoulders came into general use. The top design speed was approximately 80 km/h (50 mph) in flat country but lower design speeds could be used in hilly or mountainous terrain. Quickly convert miles into kilometres (80 miles to km) using the online calculator for metric conversions and more.What is 80 km/h in mph? How fast is 80 kilometers per hour? Instantly Convert Miles Per Hour (mph) to Kilometres Per Hour (km/h) and Many More Speed Or Velocity Conversions Online.1 Kilometer per hour (kph, km/h) 0.277 777 778 meters per second (SI base unit). Source unit: kilometers per hour (kph, km/h, kmh-1).miles per hour (mph). mile per hour. 130 kilometers per hour is 80.78 miles per hour. Quickly convert kph into mph (kph to mph) using the online calculator for metric conversions and more.80. 934 mph. 129 kph, 80. 621 mi 1 km. Both miles per hour and knots is a speed which is the number of units of distance that is covered Full mph to kph conversion tables for speeds and Roof Racks Cargo Carriers. 80 km/h 50 Mph 40 km/h 25 Mph. 80 kilometer/hour 49.709 695 379 mile/hour (mph).Careers. Answers.com WikiAnswers Categories Science Earth Sciences Natural Disasters Hurricanes Typhoons and Cyclones 80km into mph? A US passenger train that derailed, killing three people, was travelling at 80mph (130km/h) on a curve with a speed limit of 30mph, data from the trains rear engine indicates. Most comments indicate that 80-100km/h would not be unreasonable, however some pickups have had documented challenges in passing the international 70 mph (113km/h) "Moose Test". Some truck owners have also voiced concerns that they are unable to get into their top cruising gear for Das Fahrzeug sollte nicht weiter als 80 km und niemals schneller als mit 55 km/h abgeschleppt werden. It is best to tow the vehicle no farther than 80 km (50 miles), and keep the speed below 55 km/h (35 mph). This on the web one-way conversion tool converts speed and velocity units from mach ( ma ) into kilometers per hour ( km/h ) instantly online. Convert 82 Kilometers/Hour to Miles/Hour | Convert 82 km/h to mph with our conversion calculator and conversion table Boeing MD- 80 Pilot Operation Sign up, tune into the things you care about, and get updates as they happen.Find a topic youre passionate about, and jump right in. Learn the latest. Get instant insight into what people are talking about now. Miles per hour is the unit used for speed limits on roads in the United Kingdom, United States and various other nations, where it is commonlyThe Online Conversion Calculator - Converter converts Miles per hour to km per hour (mph to km/h) and kmh to mi/h (kilometers/hour to mph. Upon this the number you have entered will be converted into miles per hour, and displayed beneath the textbox. To convert mph into km/h click here. The calculated flying distance from Bratislava to Vienna is equal to 34 miles which is equal to 55 km. If you want to go by car, the driving distance between Bratislava and Vienna is 79.68 km.50 mph (80 km/h). 00 hours 59 minutes. convert 25 mph into km/h. Am I Dumping it Right? 09 Feb 2017 15:58.Things are gonna get a little sketch this weekend. As we move into the final month of summer Road speed limits are given in kilometers per hour which is abbreviated as kph or km/h.21mph. 33.80kph. Speed limits in the United States are set by each state or territory. Highway speed limits can range from an urban low of 35 mph (56 km/h) to a rural high of 85 mph (137 km/h). Speed limits are typically posted in increments of five miles per hour (8 km/h). Quickly convert miles into kilometres (80 miles to km) using the online calculator for metric conversions and more.miles/hour to kilometers per hour) and km/h to mph (kilometers/hour to miles/hour) Online Conversion Calculator - Converter. It has a range of about 80 km at 30 km/h (50 miles at 19 mph) or 25 km at 60 km/h (15.5 miles at 37 mph). Data such as current, voltage, speed, and GPS coordinates can be displayed on a linked Android smartphone, with basic functions being controlled using a 5-button control panel. This is some tractor trailer security camera footage from Swedish shipping company PostNord of a group of robbers breaking into the back of truck while traveling down the highway at 80km/h (50MPH) to take a peek at the goods inside and decide if theyre worth stealing (previously: a It is very simple multiply MPH by 1.6 you will get KM/H MPH x 1.6 KM/H. Image: Message: . A US passenger train that derailed, killing three people, was travelling at 80mph (130km/h) on a curve with a speed limit of 30mph, data from theWant to dive into a new industry? Go to a hackathon. A Couple Crazy Things Thatve Happened Recently. Definition: Mile/hour Miles per hour (symbol: MPH) is a measurement of speed in the imperial and United States customary unit.80.4672 km/h. The average human walking speed is estimated to be close to 5 kilometers per hour, and the average speed of road cyclist in the city may vary between 10 and 15 km/h.80 kph. 49.7097 mph. Supose you want to convert 80 kilometers per hour into miles per hour. In this case just write the equation, then do the mathhow much are 80 km/h in miles per hour?

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You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters? Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares? Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make? Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it. Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it. Can you work out how to balance this equaliser? You can put more than one weight on a hook. Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table? An investigation that gives you the opportunity to make and justify predictions. This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture? You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream. Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes? A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar? Find out what a "fault-free" rectangle is and try to make some of your own. How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle? Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line? Find all the numbers that can be made by adding the dots on two dice. Moira is late for school. What is the shortest route she can take from the school gates to the entrance? 10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways? These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach. Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour. How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...? This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . . Try out the lottery that is played in a far-away land. What is the chance of winning? Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up? How many different triangles can you draw on the dotty grid which each have one dot in the middle? Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume? Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once. Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own? Can you put the numbers 1 to 8 into the circles so that the four calculations are correct? Can you find all the different triangles on these peg boards, and find their angles? How will you go about finding all the jigsaw pieces that have one peg and one hole? How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways? This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code? Can you draw a square in which the perimeter is numerically equal to the area? Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles? Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make? How can you put five cereal packets together to make different shapes if you must put them face-to-face? Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had. There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules? My briefcase has a three-number combination lock, but I have forgotten the combination. I remember that there's a 3, a 5 and an 8. How many possible combinations are there to try? Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

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Maths Chapter 2 MCQ Tests Campus platform has a collection of 0 questions added to 13 chapters of Physics 10th Class. Find the degree of the polynomial 4x4 + 0x3 +0x5 + 5x+7? Download all the latest, and stay ahead of other students. Get score and analysis of your performance instantly and improvise your performance in next attempt. 2.2) – Bahupad in Hindi Medium and English Medium free to download in PDF or study online without downloading, updated for new academic session 2020-21. Online tests for class 5 Maths are available here. c) 4 and –5 Putting these values in general form of a quadratic polynomial ax2+bx+c, For any further discussion, please join the Discussion Forum. The number polynomials having zeroes as –2 and 5 is more than 3. Login/Register. p(x)=k(x2−Sx+P)where k is any real number. Prepare for various Class 10 exams at India’s leading online test portal. It will also help you in Summative Assessment and Formative Assessment examination. Solutions are applicable for all the boards which are following the Textbooks of NCERT or equivalent books. This section will really be helpful for the students of the 10th Class / SSC-2. The NCERT for Class 10 Maths Chapters wise relevant topic covered … NCERT Text Book Chapter NCERT Solutions Exemplar Solutions Objective Questions … Solution Let a is any odd positive integer number. These questions have the highest probability of coming in the examinations as per the previous year question paper pattern. Students can practice numerous questions related to Data Handling in this chapter. Prepare online for Chemistry 10th Class, 10th class Chemistry online mcq test with answers pdf,Chemistry chapter wise test On this page, you can find all online quizzes or mock tests we have to offer. Class 10 maths notes according to FBISE syllabus. Online Practice test for class 10 mathematics includes probability , statistics , quadratic equations and many more important topics. The free online mock tests for CBSE Class 3 should be used by students to check their understanding of their concepts of CBSE Class 3 Mathematics. c) none of these Then students are to inform that these tests are designed in such a way that these are just according to the pattern that you will face in the final examination. b) no real zeroes Chapter 1 Value Added Tax Chapter Test; ICSE Math Solution Class 10 Chapter 2 Banking. Sampe Paper Standard. Contact US. Maths 2 MCQ Tests Campus platform has a collection of 34 questions added to 13 chapters of Physics 10th Class. ... 4. 1/4 . Now put x = 15 in the polynomial, we get (15) 2 - 16*15 + 30 = 225 - 240 + 30 = 255 - 240 = 15. Click HEre and Download Chapter 12. Students can easily download and access the chapters of Class 10 Maths. Prepare from hundreds of questions for Chapter 2 mathematics 10th class. It will also help in testing your level of knowledge of this chapter and if you lack the understanding, you can put some extra effort. Free PDF download of Important Questions with solutions for CBSE Class 10 Maths Chapter 2 - Polynomials prepared by expert Mathematics teachers from latest edition of CBSE(NCERT) books. NCERT Text Book Chapter NCERT Solutions Exemplar Solutions Objective Questions Previous Years Solved Papers Chapter-wise QB. Product of Pragati’s age 2 years ago and years hence is 84. CBSE online test for class 10 has Multiple choice Questions for all the topics in class 10 Maths. The list of important questions for class 10 maths chapter 2 is prepared by our subject experts at Vedantu after thorough research. CBSE Class 6 Maths Chapter 10: Mensuration. You can also Download NCERT Solutions for Class 10 Maths to help you to revise the complete Syllabus and score more marks in your examinations. Question 1: A card is accidently dropped from a pack of 52 playing cards . Here on AglaSem Schools, you can access to NCERT Book Solutions in free pdf for Maths for Class 9 so that you can refer them as and when required. Become our. CBSE Class 10th Maths Assignments and Sample Papers CBSE syllabus, sample paper, unit test and assignments for mathematics for class IX and X. Free download NCERT Solutions for Class 9 Maths Chapter 10 exercise 10.1, 10.2, 10.3, 10.4, 10.5, 10.6 of Circles in PDF form. The different types of equations and their components have been described in this NCERT Maths Class 10 Chapter 2. Find the number of zeroes of p (x) in each case. Access full series of free online mock tests with answers from Mathematics Class 10. Let the number of students in class X be 100, and M and S represent the set of students who passed in math and science respectively. 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The maturation of purely mathematical mind That period before Euclid Greek and Roman mathematicians The Greeks divided mathematics into two areas, both of which had their origins in practical applications: arithmetic (the study of “multitude,” or discrete quantity) and geometry (the study of “magnitude,” or continuous number). According to Proclus’s Commentary on Euclid, ancient Egyptian surveying procedures gave rise to geometry (which literally translates to “measure of land”) because yearly flooding along the Nile prompted the Egyptians to redraw property borders. In a similar vein, it was Phoenician merchants that advanced the field of mathematics. Even though Proclus wrote his novel in the fifth century CE, his concepts may be traced back to the likes of Herodotus (mid-fifth century BCE) and Eudemus (a student of Aristotle) from even earlier periods (late 4th century BCE). If you are struggling with percentages or any other mathematical process, Visit their website to solve percentage queries. Since there is so little evidence of applied mathematics from the early Greek era, the idea is plausible but difficult to test (roughly, the 8th through the 4th century BCE). Stone tablets, for example, attest to the widespread adoption of a numerical system conceptually comparable to the widely used Roman numerals. There are perhaps a dozen abacuses made of stone that date back to the fifth and fourth centuries BCE, indicating that Herodotus was likely aware of the abacus’s use as a calculation instrument in both Greece and Egypt. When surveying new cities in Greek colonies in the sixth and fifth centuries BC, standard lengths of 70 plethra (one plethron equals 100 feet) were frequently used as the diagonal of a square of side 50 plethra; in reality, the diagonal of the square is 50Square root of2 plethra, so using 7/5 (or 1.4) as an estimate for Square root of2 is equivalent to using 7/5 (or 1.4) as an estimate for Square root of Eupalinus of Megara, an engineer from the sixth century BCE, is credited for channelling an aqueduct through a mountain on the island of Samos, although the method he used is controversial among historians. In his Laws, Plato seems to argue that the Egyptians’ method of teaching math to youngsters via practical issues from everyday life was an example the Greeks should follow. These hypotheses on the character of early Greek practical mathematics are supported by later sources, such as the arithmetic problems in papyrus writings from Ptolemaic Egypt (starting in the third century BCE) and the geometric manuals by Heron of Alexandria (1st century CE). Essentially, this Greek habit was quite close to that of ancient Egypt and Mesopotamia. There is little doubt that the Greeks borrowed ideas and concepts from much earlier civilizations. The Greeks are commonly credited as the “creators of mathematics,” yet it was the theoretical foundations of their discipline that truly set them apart. This means that mathematical statements hold true everywhere and can be proven correct. For instance, the Mesopotamians created techniques for testing whether or not the equation a2 + b2 = c2 holds for a given set of whole integers a, b, and c. (e.g., 3, 4, 5; 5, 12, 13; or 119, 120, 169). The Greeks demonstrated a general way for producing such sequences, commonly known as Pythagorean triples: for every pair of whole integers p and q, even or odd, a = (p2 + q2)/2, b = pq, and c = (p2 + q2)/2. Such numbers satisfy the relation for Pythagorean triples, as established by Euclid in Book X of the Elements. This conclusion was proved by the Greeks as part of a more comprehensive explanation of the characteristics of flat geometric forms (Euclid proves it twice, once in Book I, Proposition 47, and once in Book VI, Proposition 31). It appears that the Mesopotamians knew that right triangles contain sides that consist of sets of the numbers a, b, and c. While Euclid’s The Elements (about 300 BCE) is often seen as a seminal work in theoretical geometry, the transition from practical to theoretical mathematics may be traced back to the fifth century BCE at the earliest. Others, including Pythagoreans Archytas of Tarentum, Theaetetus of Athens, and Eudoxus of Cnidus, expanded the theoretical form of geometry by building on the foundations laid by Pythagoras of Samos (late 6th century) and Hippocrates of Chios (late 5th century) (4th century). There are no surviving copies of any of these men’s writings, therefore what we know about them comes from the opinions of other authors. While this sliver of evidence does demonstrate the extent to which Euclid relied on them, it does nothing to explain what motivated their research. Discussion centres on the how and why of this theoretical change. Commonly cited is the finding of irrational numbers. The early Pythagoreans held the concept that “all things are number” as a fundamental principle. Even though the Greek word for number, arithmos, can only be used to describe whole numbers and, in some cases, ordinary fractions, it is possible to assign a number to any geometric measure (that is, some whole number or fraction; in modern terminology, rational number). In common speech, this is commonly taken for granted, as when the length of a line is expressed as a full number of feet plus a fraction of a foot. The lines that form the square’s sides and diagonal are an exception to this rule. (For instance, assuming that the ratio of two whole integers can be stated for the side and diagonal ratios, it can be demonstrated that they must be even. Since any fraction may be expressed as the ratio of two whole numbers with no common denominator, it is obvious that this cannot occur. This has the geometric implication that no length may be used as a unit of measure for both the side and the diagonal; that is, the side and the diagonal cannot both equal the same length multiplied by (different) whole integers. This is why the Greeks used the term “incommensurable” to characterise such comparisons of lengths. (Modern mathematicians use the word “number” to refer to irrational numbers like Square root of 2, which the Greeks did not.) Although this was already widespread knowledge by the time of Plato, some late writers, such as Pappus of Alexandria (4th century CE), claim that it was discovered inside the school of Pythagoras in the 5th century BCE. By 400 BCE, it was generally accepted that lines corresponding to the square root of 3, the square root of 5, and other square roots are not directly equivalent to a standard unit of length. An much more complete discovery, that square root of p is irrational whenever p is not a rational square integer, is attributed to Plato’s companion Theaetetus. Book X, Section II, Proposition 115 of the Elements demonstrates that the effort of their pupils finally consolidated into a cohesive system, building on the foundation laid by Theaetetus and Eudoxus. The discovery of irrationals unquestionably changed the trajectory of early mathematical investigation, regardless of any assumptions made in practical practise. As the irrationals demonstrated, mathematics on its own couldn’t accomplish what geometry needed to do. All mathematical assumptions were theoretically rendered suspect once seemingly obvious ones, such as the commensurability of all lines, were revealed to be incorrect. A minimum level of justification was required for all mathematical claims. The need to identify what makes a certain chain of reasoning worthy of the label “evidence” arose as a more basic issue. Evidently before his death in the fifth century BCE, Hippocrates of Chios and his contemporaries began gathering geometric findings into textbooks dubbed “elements” (meaning “fundamental outcomes” of geometry). A century later, when Euclid was writing his comprehensive textbook, they would be among his key sources. There was fierce rivalry among the early mathematicians, who were part of a larger intellectual community that included pre-Socratic philosophers in Ionia and Italy and Sophists in Athens. Parmenides, a Greek philosopher from the fifth century BCE, challenged the basic basis of knowing when he claimed that only unchangeable objects could have actual existence. Heracleitus (c. 500 BCE) claimed, on the other hand, that the stability of our senses is an illusion created by a balance of opposing forces. Knowledge and proof both have their respective meanings questioned as a consequence. In several of the disagreements, mathematical issues served as a focal point. The Pythagoreans (and Plato, who came after them) used the certainty of mathematics as a model for deducing truths in areas such as politics and ethics. Yet others thought that mathematics was riddled with contradictions. Paradoxes about motion and quantity have been attributed to Zeno of Elea, who lived in the fifth century BCE. The premise that a line may be bisected an endless number of times gives rise to a paradox since the outcome can be either a set of points of zero length (in which case the total of an infinite number of such points is zero) or a collection of minuscule line segments (in which case the sum is infinite). In actuality, the length of the provided line must be both and. In the fifth century BCE, Democritus and other atomist philosophers attempted to answer this question by positing that everything in the cosmos is made up of infinitesimally small particles called “atoms” (from the Greek atomon, meaning “indivisible”). However, the idea of incommensurable lines in geometry ran counter to this view since atoms would then be employed to quantify all lines. Tangents to circles can be confusing; not even Sophist Protagoras and Democritus could agree on whether they meet the circle at a point or a line. During the fifth century BCE, Socratic philosophers Antiphon and Bryson grappled with the issue of equating the circle and the polygons that may be made within it. The pre-Socratics were the first to point out the problems with fundamental ideas like “existence” and “proof,” as well as more specific ones like “infinitely many” and “infinitely tiny.” These philosophical considerations may or may not have impacted mathematicians’ technical research, but they definitely made them more careful of making overly broad claims about their field’s breadth. Any examination of the possible ramifications of such circumstances is, at best, hypothetical due to the incoherence of the sources and the lack of clarity with which the mathematicians responded to the issues posed. Greek mathematics is distinctive due to its meticulous examination of fundamental assumptions and emphasis on strong proofs. While it would be impossible to offer a comprehensive analysis of the causes of these changes, we may look to the technical advances and cultural climate of the early Greek tradition as two possible explanations.

REPORT AIM The aim of this experiment is to: ? Explore the equations of uniform accelerated motion and investigate the relationship between displacement and time ? Determine the magnitude of deceleration due to friction. ? Assess the effect of mass on the car’s accelerated motion. DESIGN Hypothesis – A car moving in a straight line with a non-zero initial velocity will finally come to a rest as a result of friction, given that the car has no engine or external tractions. This motion can be considered as a uniform accelerated motion because: 1. The car is moving in a horizontal straight line so weight is cancelled by the normal reaction force from the ground. The only other force existed is the friction between the car and the surface therefore it will be equal to the net force 2. According to the formula Fr = ? N, the amount of the friction depends on two factors: the friction coefficient and the normal reaction force, both of which are fixed. Therefore the amount of friction is constant throughout the motion 3. According to Newton’s Second Law F = ma, a constant net force will result in a uniform acceleration (deceleration). The acceleration is negative in this case as cars are slowing down to a rest. For convenience, this decelerated motion can be inverted into an equivalent motion in which the car is acceleration from rest. It should follow the equation of x = ut + 1/2 at2, where x = distance traveled, u = 0 (seen as the initial velocity but actually is the final velocity which is zero at rest), a = acceleration (actually deceleration) and t = time taken during that motion. This formula can be simplified as x = 1/2 at2. We will measure the variables of x and t to verify this relationship and determine the magnitude of this deceleration, which can be derived from the gradient of the regression line of x against t2. Theoretically the mass of the car should not influence its deceleration because: Friction is the net force, Fr =? N = ? mg (as N=Weight) = Fnet = ma => ? mg = ma => a = ? g, which implies deceleration depends only on the friction coefficient. The car’s initial velocity at the beginning of the horizontal tract is provided by releasing the car from a ramp connected to the tract. This design will reduce human interventions to the car’s horizontal part of motion. Three cars of different masses will be tested to see if mass have any effect on the magnitude of deceleration. METHODS A plastic track is connected to a ramp which has a height of 9cm and contact surface of 10cm. A car is hold still on the ramp surface at a position of 3cm from the top (i. e. it will travel 7 cm to reach the bottom of the ramp). Three different cars are used in this experiment and have masses of 0. 231kg, 0. 358kg and 0. 535kg respectively. All of the three cars are released from the same position. Due to gravity, the car will gain a velocity when entering the horizontal tract. By using a stopwatch, we take measurement of the time from the moment when the car’s rear wheels first reach the horizontal tract and stop the measurement as soon as it comes to rest. The distance traveled by the car along the plastic tract is measured by a hard ruler and distance is taken from the back of the car’s rear wheels (once stopped) to the junction between the ramp and the plastic tract. [pic] This experiment is repeated three times and all data are entered into excel. A graph of distance against time squared is plotted and a regression analysis is performed to determine the relationship between the two variables and the gradient of the line. RESULTS The results of the experiment are summarized in the following table: | |Distance x (m) |Time (s) |t^2 | |Exp1 Car1 |0. 235 |0. 78 |0. 6084 | |Exp1 Car2 |0. 285 |1. 73 |2. 9929 | |Exp1 Car3 |0. 403 |2. |5. 29 | |Exp2 Car1 |0. 287 |1. 3 |1. 69 | |Exp2 Car2 |0. 308 |1. 82 |3. 3124 | |Exp2 Car3 |0. 335 |2. 03 |4. 1209 | |Exp3 Car1 |0. 104 |0. 63 |0. 3969 | |Exp3 Car2 |0. 255 |1. 54 |2. 3716 | |Exp3 Car3 |0. 374 |1. 98 |3. 9204 | Car 1 – 0. 231kg Car 2 – 0. 358kg Car 3 – 0. 535kg The Graph of distance against time squared: [pic] The gradient of the regression line is equal to 1/2 a, so a = 2 x 0. 0479 = 0. 0958 ms-2. The uncertainty for time measurement is estimated to be 10% and 1% for distance measurement (±2mm). As a = 2x/(t2), the uncertainty percentage for a is approximately = 1 % + 10% + 10% = 21%. Therefore a = 0. 0958±0. 0201 ms-2. DISCUSSION From the above graph, it is evident that there is a strong linear positive relationship between distance and time squared (R2 = 0. 8067). The distance traveled by the car during deceleration is proportional to the square of time. This is consistent with our hypothesis that x = 1/2 at2. The gradient of the regression line is equal to 1/2 a. Thus, deceleration = 2 x gradient = 2 x 0. 0479 = 0. 0958 ms-2. A R2 value of 0. 8067 means nearly 81% of the variation in distance can be explained by variation in time, according to the linear model of x against t2. This suggests that mass of the car is not a factor determining the magnitude of acceleration and does not influence the pattern of this decelerated motion. There are a few factors in this experiment which may cause variations to the result. For example, we assume the friction coefficient of the plastic tract is the same for the three cars thus they will have the same deceleration, as discussed in Experimental Design. However, the tyres of the three cars might be made from different materials so the friction coefficients might be slightly different. As a result, the deceleration is not actually constant and this creates error for our linear regression. Another compounding a factor is the joint between the ramp and the horizontal tract. As the car passes through this point, the direction of the movement may be slightly diverted due to this sharp turn. Other issues affecting experimental results include not-leveled tract, air resistance, and human errors in measurements such as delayed timing. This experiment can be improved by eliminating these compounding factors. For example, we could use a spirit level to check the plastic tract and ensure that cars are moving in a horizontal plane. The cars’ tyres should be made of the same material. The bottom of the ramp should be connected to the plastic track smoothly without any sharp corners. More measurements are to be taken to increase the accuracy of data. CONCLUSION From this experiment, we successfully verified the equation for a uniform decelerated motion x = 1/2 at2. Our results showed that distance was proportional to the square of time. The deceleration was determined from the gradient of the regression line of x against t2, which was found to be 0. 0958±0. 0201 ms-2. The mass of the car has no effect on the value of deceleration and the pattern of motion. The accuracy of the results can be improved by eliminating compounding factors such as different tyre material and tract surface layout.

Mathematics - Mathematical Analysis Integration Techniques for Rational Functions Hello its me again drifter1! Today we continue with Mathematical Analysis getting into another Integration Technique! The Integration of Rational Functions is not so difficult, but needs caution cause we can make mistakes in simple algebra calculations. You can check out my previous post about the Integration by Parts Technique here. So, without further do let's get started! Rational Cases we already know: Let's first get into Integrals that you already know how to solve, but are Rational. - Integral of f'(x)/f(x) gives us ln|f(x)|+ c and doesn't need anything more - Integral of 1/x^2+1 that gives us arctan(x) and other simple integrals like that - Integrals of 1 / x^2 +- a^2 can be found using Substitution - Integral of 1 / root(something) can be found using Substitution Using Substitution to get a "solvable" rational integral: 1. If the rational function contains e^x then we set t = e^x => x = lnt and so dx = dt/t We set t = e^x and end up with: This is a rational integral that can be solved with the techniques that we will cover in a bit. Also, don't forget that sinh(x) and cosh(x) can be represented with e^x like that: to help us solve rational integrals that contain them or even tanh(x) = sinh(x)/cosh(x). 2. If the integral contains a n-root(ax+b/cx+d), where n a natural number then we set: and end up with a rational function that contains t and can be solved with the techniques we will cover in a sec. If the integral contains n-roots of ax+b/cx+d with n = n1, n2, ..., nn then we find the least common multiple (LCM) of n1, n2, ..., nn and t will equal this LCM-root of ax+b/cx+d. Rational Function Integration Techniques: A Rational Function is a quotient of the form P(x)/Q(x) . As you already saw previously we can use substitution to get a rational integral from difficult integrals that else would be unsolvable. But, we didn't cover how to solve this rational integral that comes out. To solve such integral we use the so called Partial fraction decomposition or expansion method. Where we write the given quotient/fraction as a sum of other fractions with simpler denominators and numerators of a smaller degree in each fraction. Those fractions can be: A/(x-r)^m, where m a natural number (Ax + B) / (x^2 + px + q)^n, where n a natural number and the equation at the denominator has no (real) solutions [p^2 - 4q <0] Doing this we create factions that are a simple f'(x)/f(x) integrals. But, this can only be done when the degree of P(x) is smaller then the degree of Q(x). So we will try to convert quotients with degree of P(x) >= degree of Q(x) to the other form. So, we end up with 2 cases depending on the degree. Case 1: (degP(x) >= degQ(x)) To convert into a Case 2 quotient we have to use the polynomial division of P(x)/Q(x). To do this we have to find a remainder r(x) and quotient q(x), with 0 <= deg r(x) <= deg Q(x) . And then the polynomials will be connected using the equation: P(x) = Q(x) * q(x) + r(x) => P(x)/Q(x) = q(x) + [r(x) / Q(x)] This second one will be a integral of the second Case. The integral of q(x) will be a pretty simple linear polynomial case. Case 2: (degP(x) < degQ(x)) We have to follow the following steps: 1. Write Q(x) as a product of fractions of the type: (x - r)^m, where r is the solution of Q(x) (x^2 + px + q)^n, where the equation has no real solutions 2. For each fraction of the type (x-r)^m we write a sum in the form: A1/x-r + A2/(x-r)^2 + ... + Am/(x-r)^m, where A1, A2, ..., Am are real numbers. In the same way for each fraction of the type (x^2 + px + q)^n we write a sum in the form: (B1x + C1) / (x^2+px+q) + (B2x + C2) / (x^2+px+q)^2 + ... + (Bnx + Cn) / (x^2+px+q)^n , where Bi, Ci are real numbers and i = 1, 2, ..., n 3. Every rational function can be written as a sum of fractions as we already know from simple Algebra in school and so this sum of fractions or Partial fraction expansion will be equal to the given P(x)/Q(x). So, we have to add those unlike-quantitie fractions and the denominator will be equal either way and so we will check for which Ai, Bj, Cj etc. the numerators are equal to each other. This will give us a linear system of Ai, Bj, Cj etc. that we have to solve to calculate those values. 4. The integral of P(x)/Q(x) can then be written as a sum of those fractions having Ai, Bj, Cj calculated. Integrals of the type A / (x-r)^m are pretty simple and are a basic f'(x)/f(x) case that gives us ln|f(x)| + c. Integrals of Ax+B / (x^2 + px + q)^n are more difficult and we have to use substitution. Setting u = x + c such a integral will become simpler and can then be solved using the same basic f'(x)/f(x) case, other basic integrals (mostly arctanx) or even another substitution! Examples of Rational Integrals: 1. (x-r)^m example Because degP(x) < deg Q(x) we are in Case2 directly and can start following the steps. x^3 - 3x + 2 = (x - 1)^2*(x + 2) So, (x - 1) ^2 will become two factions A/x-1 + B/(x-1)^2 (x + 2) will get only one faction C/x+2 This means that P(x)/Q(x) = A/x-1 + B/(x-1)^2 + C/x+2 that becomes: P(x) / Q(x) = A(x-1)(x+2) + B(x+2) + C(x-1)^2 / (x+2)(x-1)^2 Where the numerator is equal to: (A+C)x^2 + (A+B-2C)x + (-2A+2B+C) We want P(x) to be equal to the numerator and so we have to solve the linear system: 4x^2= (A+C)x^2 => 4 = A + C -3x = (A+B-2C)x => -3 = A + B - 2C 5 = -2A + 2B + C Solving this system with any method you like you will end up with: A = 1, B = 2 and C = 3 And so P(x)/Q(x) = 1/x-1 + 2/(x-1)^2 + 3/x+2. Integral(P(x)/Q(x))dx = integral(1/x-1)dx + 2*integral(1/(x-1)^2)dx + 3*integral(1/x+2)dx = ln|x-1| + 2*(-1)*1/(x-1) + 3*ln|x+2| +c = ln|x-1| -2/x-1 + 3*ln|x+2| + c 2. (x^2 + px + q)^n example Because degP(x) < deg Q(x) we are again in Case2. x^2 + 2 = 0 has no solutions and so we have to write P(x)/Q(x) as a sum of fractions like that: (Ax + B) / (x^2+2) + (Cx + D) / (x^2+2)^2 If we add those fractions we end up with: P(x)/Q(x) = Ax^3 + Bx^2 + (2A+C)x + (2B+D) / (x^2+2)^2 We want P(x) = x^3 = Ax^3 + Bx^2 + (2A+C)x + (2B+D) and so: x^3 = Ax^3 => A = 1 0x^2 = Bx^2 => B = 0 0x = (2A+C)x => -2A = C => C = -2 0 = 2B + D => D = -2B => D = 0 So, A = 1, B = D = 0 and C = -2. Setting those values on our fraction sum we end up with: P(x)/Q(x) = x/(x^2+2) + -2x/(x^2+2)^2. Integral(P(x)/Q(x))dx = integral(x/(x^2+2))dx + integral(-2x/(x^2+2)^2)dx The first one is again a ln|f(x)| case and the second again a power case and so the result is: 1/2*ln(x^2+2) + 1/(x^2+2) + c 3. combined types of fractions You can directly see that P(x) < Q(x). x^4 - 1 = (x-1)(x+1)(x^2 + 1) So, we have two types of fractions and 1/x^4-1 now looks like this: A/x-1 + B/x+1 + Cx+D/(x^2+1) = (A+B+C)x^3 + (A-B+D)x^2 + (A+B-C)x + (A-B-D) / x^4-1 If we set 1 equal to the numerator we end up with the following system: A+B+C = 0 A-B+D = 0 A+B-C = 0 A-B-D = 1 If you solve this system using any method you like you will end up with: A = 1/4, B = -1/4, C = 0, D = -1/2 And so our integral now looks like this: integral(P(x)/Q(x))dx = 1/4*integral(1/x-1)dx -1/4*integral(1/x+1)dx -1/2*integral(1/x^2+1). The first two are simple ln|f(x)| cases, but the last one is interesting cause its actually arctan(x)! So, our final result is: 1/4*ln|x-1| -1/4*ln|x+1| - 1/2*arctan(x) + c 4. In Case1 degP(x)>degQ(x) and so we are in Case1 Knowing how to divide polynomials you do the following: This explains them pretty good if you don't know them already. This means that we know have to solve the integral: x^2+3x -1 + (-2x+1)/x^2+1 We know that x^2+1 has no solution and so we can't tranform it into a Case 2. This means that we will try to create simple integrals in another way. We will simply rewrite the numerator as -2x +1 = -(x^2+1)' + 1 and solve the simple ln|f(x)| case and create a simple arctan(x) case. So, our final integral is: Integral(P(x)/Q(x)) = integral(x^2+3x-1)dx - integral(x^2+1)'/(x^2+1))dx + integral(1/x^2+1)dx = x^3/3 + 3x^2/2 - x - ln(x^2+1) + arctan(x) + c . Try solving the e^x integral example that we didn't finished with the function that contains t. You must get ln|t^2 - 25| + c = ln|e^(2x) - 25| + c And this is actually today's post and I hope you enjoyed it! The next and final techniques that I will cover in our series will be about solving integrals of trigonometric functions. After that we will get into how we solve Limits that contain Roots, something that I completely forgot to talk about my Limit posts.

In the visible spectrum which color has the shortest wavelength Sauce financiere pour bouchees a la reine Violet. Violet waves have the most energy of the visible spectrum. Remember: " "c=flambda Therefore: " "f=c/lambda Here c is the speed of light in a vacuum. So: As wavelength decreases, frequency increases and, as E=hf, where h is constant (Planck's constant), so does the energy that the waves carry. Waves with a short wavelength have the most energy. Red waves have a relatively long ...|Light velocity v = 3×108m/sec. According to the light wavelength formula λ= ν*f. λ = (3×10^8 / 1)* 06.24×1014. λ = 4.80 x 10−7. Thus, this is all about the wavelength of visible light. From the above information, finally, we can conclude that these light waves are electromagnetic waves that are visible.| which part color of visible light has the longest wavelength || Shortest wavelength: Violet In the visible spectrum the shortest wavelength is of violet. Violet is right before ultraviolet in the spectrum. And according to the wave equation (i.e. speed= frequency * wavelength), frequency is inversely proportional to wavelength so shortest …|As the full spectrum of visible light travels through a prism, the wavelengths separate into the colors of the rainbow because each color is a different wavelength. Violet has the shortest wavelength , at around 380 nanometers, and red has the longest wavelength , at around 700 nanometers.|True or false: as you move along the spectrum, left to right, the wavelengths decrease in size (get smaller) Electromagnetic Spectrum Test Review DRAFT. 6th grade. 112 times. Other Sciences. ... Q. Put the visible light colors in order from longest wavelength to shortest wavelength. answer choices . Violet, Indigo, Blue, Green, Orange, Yellow ...| Visible light. Visible light is the small part within the electromagnetic spectrum that human eyes are sensitive to and can detect. Visible light waves consist of different wavelengths. The colour of visible light depends on its wavelength. These wavelengths range from 700 nm at the red end of the spectrum to 400 nm at the violet end.| Also, what color has the shortest wavelength? As the full spectrum of visible light travels through a prism, the wavelengths separate into the colors of the rainbow because each color is a different wavelength. Violet has the shortest wavelength, at around 380 nanometers, and red has the longest wavelength, at around 700 nanometers.| Visible light has a wavelength range from ~400 nm to ~700 nm. Violet light has a wavelength of ~400 nm, and a frequency of ~7.5*10 14 Hz. Red light has a wavelength of ~700 nm, and a frequency of ~4.3*10 14 Hz. Visible light makes up just a small part of the full electromagnetic spectrum.|Violet is the most energetic color and red is the least. According to the figure, if someone shined light with a wavelength of 550 nm at us it would look green. If someone shined white light at us, what wavelength does it have? White is not in our visible spectrum because it is composed of all the wavelengths of light. A light bulb is a good ...| The shortest wavelength of visible light produces the color. The shortest wavelength of visible light quizlet. The shortest ... , the concept of the visible spectrum has become more defined, as the light out of the visible range has been discovered and characterized by William Herschel (infrared) and Johann Wilhelm Ritter (Ultravioletto ...| The first factor, hue is what we are usually talking about when we refer to color (a red shirt has a red hue). The hue is basically the specific name for the specific wavelength that is reflected by the object. Violet has the shortest visible wavelength in the visible spectrum (~ 400 nm), and red has the longest (700 nm).User: The shortest wavelength within the visible spectrum is _____ light. A. red B. blue C. violet D. orange Weegy: The shortest wavelength within the visible spectrum is violet. Score 1 User: If a person looking at a poster sees green instead of yellow and doesn't see red at all, this person most likely has color blindness where _____ nerves fail to respond to light properly.|Which color of visible light has the shortest wavelength? the longest wavelength? the lowest frequency? the highest frequency? What color of light in the visible spectrum appears brightest Asked by wiki @ 12/06/2021 in Physics viewed by 31 persons|Gamma rays, a form of nuclear and cosmic EM radiation, can have the highest frequencies and, hence, the highest photon energies in the EM spectrum.For example, a γ-ray photon with f = 10 21 Hz has an energy E = hf = 6.63 × 10 −13 J = 4.14 MeV. This is sufficient energy to ionize thousands of atoms and molecules, since only 10 to 1000 eV are needed per ionization.|lengths of visible light as red and the shortest as violet. This narrow band is very small compared with the rest of the spectrum. In fact, visible light is only about 1/100,000 of the complete EM spectrum. The area below visible light and above microwaves is the infrared part of the EM spectrum. Above visible light is the ultraviolet part of ... |The visible light spectrum (380−750 nm) is the light we are able to see. This spectrum is often referred to as "ROY G BIV" as a mnemonic device for the order of colors it produces. Violet has the shortest wavelength (about 400 nm) and red has the longest wavelength (about 650-700 nm).|The electromagnetic spectrum is a range of frequencies of different energy waves such as gamma rays, X rays, ultraviolet rays, visible light, infrared waves, microwaves and radio waves. The visible light frequencies lie between the frequencies of the ultraviolet rays and infrared waves. Color. Frequency (THz) Wavelength (nm) Red. 400-484. 620-750.|The color with the shortest wavelength among all colors in light spectrum is violet, in electromagnetic spectrum it is the gamma ray with the shortest wavelength among other waves . THE LIGHT AND ELECTROMAGNETIC SPECTRUM. Light spectrum refers to the range of colors, ...|WAVELENGTHS OF VISIBLE LIGHT. As the full spectrum of visible light travels through a prism, the wavelengths separate into the colors of the rainbow because each color is a different wavelength. Violet has the shortest wavelength, at around 380 nanometers, and red has the longest wavelength, at around 700 nanometers.

3 edition of Applied Derivatives found in the catalog. January 15, 2002 by Blackwell Publishers Written in English |The Physical Object| |Number of Pages||384| Applied Calculus Math Karl Heinz Dovermann Professor of Mathematics University of Hawaii derivative is at least as clear in our approach as it is in the one using limits. and a high school level book by M¨uller which use this approach. Calculus was File Size: 1MB. The Definition of the Derivative – In this section we will be looking at the definition of the derivative. Interpretation of the Derivative – Here we will take a quick look at some interpretations of the derivative. Differentiation Formulas – Here we will start introducing some of the differentiation formulas used in File Size: 2MB. The book is in use at Whitman College and is occasionally updated to correct errors and add new material. The latest versions may be found by going to this work or a derivative, include the history of the document. This text was initially written by David Guichard. The . Partial Derivatives, pp. Surface and Level Curves, pp. Partial Derivatives, pp. Tangent Planes and Linear Approximations, pp. Directional Derivatives and Gradients, pp. The Chain Rule, pp. Maxima, Minima, and Saddle Points, pp. Finding increasing interval given the derivative (Opens a modal) Increasing & decreasing intervals review (Opens a modal) Rates of change in other applied contexts (non-motion problems) 4 questions. Practice. Mean value theorem. Learn. Derivative applications challenge. 4 questions. Practice. Derivatives describe the rate of change of quantities. This becomes very useful when solving various problems that are related to rates of change in applied, real-world, situations. Also learn how to apply derivatives to approximate function values and find limits using L’Hôpital’s rule. Adventures of Thumbelina Isolated perfused organ preparations Monument to the memory of General Andrew Jackson Politics of Sino-Indian confrontation. Storage of fall-harvested potatoes in the northeastern late summer crop area Local wage rates for selected occupations in public and private construction, 1936. Inaugural address of Robert Maynard Hutchins Technical report on the potential growth of obnoxious aquatic plants and practical systems of control in the Republic of India Report of all contracts for carrying the mail made within the fiscal year ended June 30, 1890. Fool of Hearts Cian and Ethne Applied Derivatives provides a detailed, yet relatively non-technical, treatment of the conceptual foundations of derivative securities markets' pricing and investment principles. This book draws from the most fundamental concepts of pricing for options, futures, and swaps to provide insight into the potential risks and returns from conventional option investing.5/5(1). Applied Math for derivatives offers a guide to the economics and valuation of financial derivative instruments which does not require a math degree to understand. It is deliberately targeted at those practitioners and students who wish to move beyond the algebra to the actual implementation of pricing and valuation models - often the difficult part of any derivative modelling by: 3. Applied Derivatives provides a detailed, yet relatively non-technical, treatment of the conceptual foundations for derivative securities markets pricing and investment principles. This book draws from the most fundamental concepts of pricing for options, futures, and swaps to provide insight into the potential risks and returns from conventional option investing. "The Applied Derivatives book and most inclusive book ever written about derivatives - a necessary reference for serious derivatives students." - Mark Rubinstein, Paul Stephens Professor of Applied Investment Analysis, University of California at Berkeley. "Likely to become the bible of financial engineering."Cited by: Applied Quantitative Finance for Equity Derivatives 1st Edition by Jherek Healy (Author) out of 5 stars 2 ratings. ISBN ISBN Why is ISBN important. ISBN. This bar-code number lets you verify that you're getting exactly the right version or edition of a book. 5/5(2). Note: If you're looking for a free download links of Applied Derivatives: Options, Futures, and Swaps Pdf, epub, docx and torrent then this site is not for you. only do ebook promotions online and we does not distribute any free download of ebook on this site. Description. Applied Derivatives provides a detailed, yet relatively non-technical, treatment of the conceptual foundations of derivative securities markets' pricing and investment principles. This book draws from the most fundamental concepts of pricing for options, futures, and swaps to provide insight into the potential risks and returns from conventional option investing. Applied Derivatives – a member of The JSE Securities Exchange South Africa- is a leading securities trading and brokerage firm. The company offers execution and structuring services based on listed derivatives; including soft commodities, financial indices, single stocks and foreign exchange. About this book. Robert Whaley has more than twenty-five years of experience in the world of finance, and with this book he shares his hard-won knowledge in the field of derivatives with you. Divided into ten information-packed parts, Derivatives shows. In this chapter we will cover many of the major applications of derivatives. Applications included are determining absolute and relative minimum and maximum function values (both with and without constraints), sketching the graph of a function without using a computational aid, determining the Linear Approximation of a function, L’Hospital’s Rule (allowing us to compute some limits we. Applied Probabilistic Calculus for Financial Engineering: An Introduction Using R provides R recipes for asset allocation and portfolio optimization problems. It begins by introducing all the necessary probabilistic and statistical foundations, before moving on to topics related to asset allocation and portfolio optimization with R codes illustrated for various by: 2. A practical, informative guide to derivatives in the real world. Derivatives is an exposition on investments, guiding you from the basic concepts, strategies, and fundamentals to a more detailed understanding of the advanced strategies and models. As part of Bloomberg Financial's three part series on securities, Derivatives focuses on derivative securities and the functionality of the. Suggested Books for MBA Financial Derivatives. Gupta S.L., FINANCIAL DERIVATIVES THEORY, CONCEPTS AND PROBLEMS PHI, Delhi, Kumar S.S.S. FINANCIAL DERIVATIVES, PHI, New Delhi, Achievements and Prospects,’’ Journal of Applied Corporate Finance, 4 (Winter ): 4– MBA 4th Sem Notes, Study Materials & : Daily Exams. In Commodity Derivatives: Markets and Applications, Neil Schofield provides a complete and accessible reference for anyone working in, or studying commodity markets and their associated derivatives. Dealing primarily with over the counter structures, the book provides extensive coverage of both hard and soft commodities, including gold, crude oil, electricity, plastics, emissions and. Get this from a library. Applied derivatives: options, futures, and swaps. [Richard J Rendleman, Jr.] -- Based on some of the ground-breaking work Richard Rendleman did helping to develop the Binomial Option Pricing Model inthis book is the culmination of 18 years of research in option pricing. Prelude to Applications of Derivatives A rocket launch involves two related quantities that change over time. Being able to solve this type of problem is just one application of derivatives introduced in this chapter. We also look at how derivatives are used to find maximum and minimum values of functions. However, the book will also be useful for applied scientists from engineering and physics.” (Jan Lovíšek, zbMATH, Vol.) “The book under review concerns new methods of solving a class of shape optimization problems appearing in continuum mechanics, mainly in solid mechanics, composites and plate-like bodies. The resulting derivative values are useful for all scientific computations that are based on linear, quadratic, or higher order approximations to nonlinear scalar or vector functions. AD has been applied in particular to optimization, parameter identification, nonlinear equation solving, the numerical integration of differential equations, and. BASICS OF EQUITY DERIVATIVES CONTENTS 1. Introduction to Derivatives 1 - 9 2. Market Index 10 - 17 3. Futures and Options 18 - 33 4. Trading, Clearing and Settlement 34 - Historical notes. In applied mathematics and mathematical analysis, a fractional derivative is a derivative of any arbitrary order, real or complex. Its first appearance is in a letter written to Guillaume de l'Hôpital by Gottfried Wilhelm Leibniz in Fractional calculus was introduced in one of Niels Henrik Abel’s early papers where all the elements can be found: the idea of. Following the table of contents in Applied Calculus 7e by Stefan Waner and Steven R. Costenoble You can get back here from anywhere by using the Everything for Applied Calc link. Note: To change the edition of the book, use the navigation on the top left.Applications of the Derivative tion Optimiza Many important applied problems involve finding the best way to accomplish some task. Often this involves finding the maximum or minimum value of some function: the minimum time to make a certain journey, the minimum cost for doing a task, the maximum power that can be generated by a device. Derivatives Demystified follows a sequence that is designed to show that, although there are many applications of derivatives, there are only a small number of basic building blocks, namely forwards and futures, swaps and options. The book shows how each building block is applied to different markets and to the solution of various risk.

COMPUTING IN STATISTICS COMPUTING IN STATISTICS STAT 517 Popular in Course Mr. Cleve MacGyver verified elite notetaker Popular in Statistics This 10 page Class Notes was uploaded by Shane Marks on Monday October 26, 2015. The Class Notes belongs to STAT 517 at University of South Carolina - Columbia taught by D. Hitchcock in Fall. Since its upload, it has received 47 views. For similar materials see /class/229653/stat-517-university-of-south-carolina-columbia in Statistics at University of South Carolina - Columbia. Reviews for COMPUTING IN STATISTICS Report this Material What is Karma? Karma is the currency of StudySoup. Date Created: 10/26/15 STAT 517 DelwicheSlaughter Chapter 2 HitchcockGrego Chapter 2 Getting Data Into SAS 0 Data stored in many different formsformats 0 Four categories of methods to read in data 1 Entering data directly through keyboard small data sets 2 Creating SAS data sets from raw data files 3 Converting other software s data files eg Excel into SAS data sets my favorite 4 Reading other software s data files directly often need additional SASACCESS products University of South Carolina Page 1 STAT 517 DelwicheSlaughter Chapter 2 HitchcockGrego Import Window 0 Allows you to import various types of data files Microsoft Excel formats 0 Default is for first row to be variable names Change this using Options button 0 Options button also selects worksheet from workbook 0 Work library data set deleted after exiting SAS 0 Other libraries data set saved after exiting SAS but not necessarily library location 0 Can save PROC IMPORT statements used to import the data University of South Carolina Page 2 STAT 517 DelwicheSlaughter Chapter 2 HitchcockGrego Reading in Raw Data If you type data directly into a SAS program this is indicated with a statement like cards datalines lines o If your raw data is in an external file use an INFILE statement to tell SAS where it is o Specify full path name 0 If your lines are longer than 256 characters use LRECL University of South Carolina Page 3 STAT 517 DelwicheSlaughter Chapter 2 HitchcockGrego Data Separated by Spaces 0 This style is called free format since the number of spaces in between variables is flexible 0 Use INPUT statement to name variables 0 Include a after names of character variables University of South Carolina Page 4 STAT 517 DelwicheSlaughter Chapter 2 HitchcockGrego Data Arranged in Columns 0 Knowledge of this approach is less important nowadays 0 Important applications still exist 0 Each value of a variable is found at the same spot on the data line 0 Advantages 1 Don t need space between values 2 Missing values don t need special symbol can be blank 3 Character data can have blanks 4 Can skip variables you don t need to read into SAS Example INPUT varl 1 10 var2 11 15 var3 16 30 University of South Carolina Page 5 STAT 517 DelwicheSlaughter Chapter 2 HitchcockGrego Data Not in Standard Format 0 Types of nonstandard data 1 Numbers with commas or dollar signs 2 Dates and times of day 0 We can read nonstandard data using codes known as informats 0 Most informats end in so SAS won t confuse them with a variable 0 Import from Excel often assigns informats automatically 0 p 4445 lists many SAS informats Note that date informats are converted to a numerical value Julian date University of South Carolina Page 6 STAT 517 DelwicheSlaughter Chapter 2 HitchcockGrego Other Inputting Issues 0 You can mix input styles read in some variables liststyle others columnstyle others using informats even the order can be shuffled o Eg you can explicitly move SAS to a specific column number Example 5 0 moves SAS to the 50th column Messy Data 0 colon modifier Tells SAS exactly how many columns long a variable s field is but stops when it reaches a space 0 Example Deptname 15 tells SAS to read Deptname for 15 characters or until it reaches a space 0 This method is not appropriate for character data with embedded spaces University of South Carolina Page 7 STAT 517 DelwicheSlaughter Chapter 2 HitchcockGrego Multiple Lines of Data per Observation 0 Sometimes each observation will be on several lines in the raw data file census data standardized test scores etc 0 Use to tell SAS when to go to the next line 0 Or use 2 for example to tell SAS to go to the 2nd line of the observation Multiple Observations per Line of Raw Data 0 Sometimes several observations will be on one line of data 0 This is common for textbook exercises 0 Use to tell SAS to stay on the raw data line and wait for the next observation Reading Part of a Data File 0 Sometimes we want to modify data input based on values of one variable 0 We can read just the first variables using the sign University of South Carolina Page 8 STAT 517 DelwicheSlaughter Chapter 2 HitchcockGrego Reading Delimited Files 0 These instructions have been completely subsumed by Excel imports o DLM allows you to have something other than spaces separated data values 0 Comma delimiters DLM 0 Tab delimiters DLM O 9 X o delimiters DLM o This assumes two delimiters in a row is the same as a single delimiter o What if two commas in a row indicate a missing value c What if some data values contain commas 0 Can use DSD option 0 Note Data values with commas in them must be in quotes 0 Default with DSD is comma delimiters but can specify other delimiters with DLM option University of South Carolina Page 9 STAT 517 DelwicheSlaughter Chapter 2 HitchcockGrego SAS data sets Temporary and Permanent 0 Data sets stored in Work library are temporary removed upon exiting SAS 0 Data sets stored in other libraries are permanent will be saved upon exiting SAS 0 You can specify the library when creating a data set in the DATA step Example Suppose you have a library called sportlib this is a libref DATA sportlibbaseball creates a data set baseball to be stored in the sport lib library permanent DATA workbaseball would store baseball in the work library temporary DATA baseball by default stores ba s ebal l in the we rk library temporary University of South Carolina Page 10

Return to Metric Table of Contents Metric conversion where only one unit is converted Go to 10 two-unit metric problems Doing this type of problem is simply a succession of conversions from one unit to another. You first convert one side of the fraction, say, the numerator, then you do the denominator. Often, a teacher will present these solutions as one long string of conversions. You also see this type of presentation in textbooks. It can be quite confusing when you first see it. This technique is called "dimensional analysis" (the older term), with "factor label method" being the newer term. DA can also called "unitary conversions," or "unitary rates." The word unitary comes from the fact that the numerator and the denominator in a conversion both describe the same quantity. As an example, take this conversion factor: 1000 mL / 1 L Both 1000 mL and 1 L describe the same-sized volume, so 1000 mL / 1 L is referred to as a unitary rate. Since 1000 mL and 1 L describe the same volume, we can think of 1000 mL / 1 L as being like multiplying some number by 1. The description of the volume changes units, but it still describes the same sized volume. The ChemTeam tends to present two-unit conversion problems as a sequence of one-step calculations. However, I will also reference the one-line type presentation that is the hallmark of dimensional analysis. On a professional basis, I do not believe the one-line approach is the proper tool to use when teaching these types of problems. There are those that disagree with me. Doing DA problems, to me, are like balancing equations or predicting products of a reaction. There are LOTS of little bits that you have to remember and, when that is the case, experience is really, really important. The problem you would face is to be able to read a one-line solution and back-track to the logic the writer used. That can be difficult, especially for a rookie. Conclusion: lots of examples for you to study! Example #1: Convert the speed of light (3.00 x 108 m/s) to km/year. 1) We'll start with the numerator, since that can be done in a one step conversion. 3.00 x 108 m 1 km –––––––––– x –––––– = 3.00 x 105 m/s 1 s 1000 m 2) Now, we turn to converting seconds to years. This I will do in a step-by-step manner. I happen to have memorized that there are 3600 seconds in one hour. So, we start with that conversion. 3.00 x 105 km 3600 s ––––––––––– x ––––– = 1.08 x 109 km/hr 1 s 1 hr 3) Continuing the calculations, we move step-by-step from hours to days and then to years (we can skip months, since we know how many days there are in a year. 1.08 x 109 km 24 hr ––––––––––– x ––––– = 2.592 x 1010 km/day 1 hr 1 day 2.592 x 1010 km 365.25 day ––––––––––––– x ––––––––– = 9.47 x 1012 km/yr (to three sig figs) 1 day 1 yr 4) If I were to present it as a one-line type calculation (the usual presentation form in dimensional analysis), it would be this: 3.00 x 108 m 1 km 3600 s 24 hr 365.25 day –––––––––– x ––––––– x ––––––– x ––––––– x ––––––– = 9.47 x 1012 km/yr 1 s 1000 m 1 hr 1 day 1 yr On the Internet, it may also be seen this way: 3.00 x 108 m/s x (1 km / 1000 m) x (3600 s / hr) x (24 hr / day) x (365.25 day / yr) = 9.47 x 1012 km/yr 5) One advantage to the above presentation is that you simply carry out the steps in sequence (divide by 1000, multiply by 3600, mult by 24 and mult by 365.25) on your calculator and then round off. 6) Doing it step-by-step results in intermediate answers along the way, but I think it's better to teach the steps rather than confront a student with the dimensional analysis method right from the start of instruction. Notes on variations of the above problem: 1) Notice that I used 365.25 days rather than 365. Using the latter figure results in an answer of 9.4608 x 1012 km/yr, which rounds off to 9.46 x 1012 km/yr. 2) This problem can start with cm/s rather than m/s. The speed of light in cm/s is 3.00 x 1010 cm/s. 3) Often, this problem ends in km/hr. Consequently, the question you could see might ask for the conversion from cm/s to km/hr. As in, right now . . . . Example #2: Light travels at a speed of 3.00 x 1010 cm/s. What is the speed of light in kilometers/hours? 1) Convert cm/s to km/s: (3.00 x 1010 cm/s) (1 m / 100 cm) (1 km / 1000 m) = 3.00 x 105 km/s 2) Convert seconds to hours: (3.00 x 105 km/s) (60 s / 1 min) (60 min / 1 hr) = 1.08 x 109 km/hr 3) A slightly more compact version: (3.00 x 1010 cm/s) (1 km / 105 cm) = 3.00 x 105 km/s (3.00 x 105 km/s) (3600 s / 1 hr) = 1.08 x 109 km/hr 4) Done in DA (dimensional analysis) style: 3.00 x 1010 cm 1 m 1 km 60 s 60 min –––––––––– x ––––––– x ––––––– x ––––––– x ––––––– = 1.08 x 109 km/hr 1 s 100 cm 1000 m 1 min 1 hr The two length conversions could be combined (1 km = 105 cm) and the two time conversions could be combined (1 hr = 3600 s). Example #3: Convert 6.43 g/mL to its equivalent in kg/L. 1) Convert grams to kilograms: 6.43 g/mL x (1 kg/1000 g) = 0.00643 kg/mL 2) Convert mL to L: 0.0643 kg/mL x (1000 mL/L) = 6.43 kg/L 3) Dimensional analysis: 6.43 g 1 kg 1000 mL –––––– x –––––– x ––––––– = 6.43 kg/L 1 mL 1000 g 1 L Comment: teachers like to ask this question on the test. Example #4: A cylindrical piece of metal is 4.50 dm in height with radius of 5.50 x 10¯5 km. (a) Calculate the volume in milliliters to the correct significant figures given V = π r2 h for a cylinder. (b) Calculate the volume in mm3 Solution to (a): 1) The key to solving part (a) is to remember that cm3 and mL are the same volume, so 1 cm3 = 1 mL. So, convert both measurements of the cylinder to cm:: 4.50 dm 10 cm ––––––– x ––––– = 45.0 cm 1 1 dm 5.50 x 10¯5 km 105 cm ––––––––––– x ––––– = 5.50 cm 1 1 km 2) Plug our numbers into the volume formula provided to get cm3. V = π r2 h V = (3.14159) (5.50 cm)2 (45.0 cm) V = 4276.5 cm3 Rounding to three sig figs and noting that 1 cm3 = 1 mL, we have this for the final answer: V = 4280 mL = 4.28 x 103 mL Solution to (b): 1) The unit we need on the height and radius is mm, so convert 45.0 cm to 450 mm and 5.50 cm to 55.0 mm. Then plug back into the volume formula: V = (3.14159) (55.0 mm)2 (450 mm) V = 4276489 mm3 To three sig figs, we have 4.28 x 106 mm3 2) Notice that both numbers got increased by a factor of 10 and then within the volume formula, there is a total factor increase of 103 (because one of factor of 10 increase was squared to give a factor of 100 increase). That means the answer to part (b) is the answer to part (a) times 1000, resulting in 4.28 x 106 mm3. Example #5: Convert 4.09 x 10¯6 kg/L to mg/cm3 using dimensional analysis. 1) When dimensional analysis is specified in a problem, the usual answer desired is in the form of all the conversions gathered together into one line. I will build the final answer up one conversion at a time. Each comment with an arrow is about the last conversion in each line. 4.09 x 10¯6 kg/L x (1000 g / kg) <--- converts kg to g 4.09 x 10¯6 kg/L x (1000 g / kg) x (1000 mg / 1 g) <--- converts g to mg 4.09 x 10¯6 kg/L x (1000 g / kg) x (1000 mg / 1 g) x (1 L / 1000 mL) <--- converts L to mL 4.09 x 10¯6 kg/L x (1000 g / kg) x (1000 mg / 1 g) x (1 L / 1000 mL) x (1 cm3/mL) <--- converts mL to cm3 the answer is 0.00409 mg/cm3 2) Notice that the above conversion converted through the base unit of grams, as in kg to g, then g to mg. You can combine those two conversions if so desired: 4.09 x 10¯6 kg/L x (106 mg / kg) x (1 L / 1000 mL) x (1 cm3/mL) = 0.00409 mg/cm3 3) Here's the most-common way DA solutions are formatted: 4.09 x 10¯6 kg 106 mg 1 L 1 cm3 ––––––––––– x –––––– x ––––––– x ––––– = 0.00409 mg/cm3 1 L 1 kg 1000 mL 1 mL 4) Some teachers prefer the DA method for homework and test answers. Some prefer the steps to be separated (with intermediate answers shown). Others do not care. Be sure to check what your teacher desires. Example #6a: Convert 303.0 mi/hr to feet/min. V = (303.0 mi/hr) (1 hr / 60 min) (5280 ft / mile) = 26600 ft/min 303 mi 1 hr 5280 ft –––––– x ––––– x –––––– = 26660 ft/min (to 4 sig figs) 1 hr 60 min 1 mi The hr/min factor converts 303.0 mi per hr to 5.05 mi per min. The ft/mi factor converts 5.05 mi per min to 26664 ft per min. The factors used algebraically cancel units to give the units wanted. The final answer is 26660 ft/s. It has been rounded off to four significant figures. Note that this example uses English units. The principles of converting are the same as with metric units. The hr/min conversion as well as the foot/mile conversion are defined amounts. As such, they play no role in determining significant figures. Example #6b: Convert 303.0 mi/hr to feet/second. The dimensional analysis set-up will be presented without comment. 303 mi 1 hr 1 min 5280 ft ––––––– x ––––––– x ––––––– x ––––––– = 444.4 ft/s <--- 4 sig figs 1 hr 60 min 60 sec 1 mi Example #7: Convert 2113 km/h into cm/s. 1) Do the hour to second conversion: 2113 km 1 hr ––––––– x ––––– = 0.5869444 km/s 1 hr 3600 s 2) Convert km to m, then convert m to cm (as opposed to converting km directly to cm in one step): 0.5869444 km 1000 m ––––––––––– x –––––– = 586.9444 m/s s 1 km 3) Now, the m to cm conversion: 586.9444 m 100 cm –––––––––– x –––––– = 58694.44 cm/s = 5.869 x 104 cm/s (to 4 sig figs) s 1 m 4) Here's the full conversion, with km being directly converted to cm: 2113 km 1 hr 105 cm ––––––– x ––––– x ––––– = 5.869 x 104 cm/s 1 hr 3600 s 1 km Example #8: How many grams of lead are there in a lead brick 5.00 cm by 13.0 cm by 24.0 cm? The density of lead is 11300 kg/m3. We could change the cm to m and calculate the volume in m3, then use the density to get the mass of lead. Another path would be to change the density to use cm3 and then calculate the volume in cm3 and thence to the mass. I think I will do both!! Solution where cm is changed to m first: 1) Change cm to m: (5.00 cm) (1 m / 100 cm) = 0.0500 m (13.0 cm) (1 m / 100 cm) = 0.130 m (24.0 cm) (1 m / 100 cm) = 0.240 m 2) Calculate the density in m3: (0.0500 m) (0.130 m) (0.240 m) = 0.00156 m3 3) Determine mass in kg, then g: (11300 kg/m3) (0.00156 m3) = 17.628 kg (17.628 kg) (1000 g / kg) = 17628 g to three sig figs, 17600 g Solution where m3 is changed to cm3 first: 1) Convert density to g/cm3: 11300 kg 1 m3 1000 g ––––––– x ––––––– x –––––– = 11.3 g/cm3 m3 (100 cm)3 1 kg Notice that I included both conversions into this step. 2) Determine volume of the lead brick: (5.00 cm) (13.0 cm) (24.0 cm) = 1560 cm3 3) Determine mass in grams: (11.3 g/cm3) (1560 cm3) = 17628 g To three sig figs, 17600 g Bonus Example: The SI unit for density is kg/m3. Convert the density of platinum (21450 kg/m3) to the more commonly-used unit of g/cm3 1) Convert kg/m3 to g/m3: (21450 kg/m3) (1000 g / 1 kg) = 21450000 g/m3 2) Convert g/m3 to g/cm3 (21450000 kg/m3) (1 m3 / 1003 cm3) = 21.45 g/cm3 Note the use of 1003. 1 m3 is a cube 100 cm on a side: 100 cm x 100 cm x 100 cm = 1003 cm3. 3) Many teachers that teach dimensional analysis want the solution in one line of calculation steps: (21450 kg/m3) (1000 g / 1 kg) (1 m3 / 1003 cm3) = 21.45 g/cm3 Note the interim values/units such as 21450000 g/m3 do not appear in a one-line dimensional analysis presentation. Textbooks will often present a dimensional analysis set-up in this manner: 21450 kg 1000 g 1 m3 ––––––– x ––––––– x ––––––– = 21.45 g/cm3 1 m3 1 kg 1003 cm3 Your teacher may require it in that manner as well. One of the advantages to the above set-up is that it's much more obvious which units cancel. For example, you can clearly see the kg in the numerator of the first factor and in the denominator of the second factor. Comment: it is easy to imagine a situation (test or homework) where, in the problem, you are given the density of a substance in units of kg/m3 but, in the problem solution, you must use the density in units of g/cm3. Consequently, I recommend that the above conversion be in your "bag of tricks." By the way, please notice that the net effect of the above conversion is to divide the kg/m3 value by 1000 to get the g/cm3 value. if you are not required to show the conversion as I did above, you can use the 'divide by 1000" step as a convenient shortcut. Go to 10 two-unit metric problems Metric conversion where only one unit is converted Return to Metric Table of Contents

Two-photon interference with independent classical sources, in which superposition of two indistinguishable two-photon paths plays a key role, is of limited visibility with a maximum value of 50%. By using a random-phase grating to modulate the wavefront of a coherent light, we introduce superposition of multiple indistinguishable two-photon paths, which enhances the two-photon interference effect with a signature of visibility exceeding 50%. The result shows the importance of phase control in the control of high-order coherence of classical light. © 2013 OSA Interference is an essentially important topic in optical physics, resulting in many interesting phenomena and important applications. The key physics lying behind interference is the superposition principle. After the birth of quantum physics, it was realized that superposition of multiple single-photon paths plays a key role in the traditional optical interference phenomenon, which is usually known as Dirac’s famous statement: “Each photon interferes only with itself. Interference between different photons never occurs”. In 1956, Hanbury Brown and Twiss (HBT) introduced the second-order correlation measurement, and reported a new type of interference effect between independent photons, i.e., the bunching effect of thermal light [2, 3]. Soon after, it was realized that superposition of two indistinguishable two-photon paths plays a key role in the HBT interferometer . To explain it briefly, as shown in Fig. 1, for every pair of independent photons, there are two indistinguishable paths for the pair of photons to trigger a coincidence count. The phase of the complex amplitude of a two-photon path is composed of two components: the initial random phase component ϕs1 + ϕs2 with ϕsi being the initial random phase of the point source emitting photon si (i = 1, 2), and the propagation phase component related to the optical paths for the photons propagating from the sources to the respective detectors . Since the amplitudes of the two paths are always of the same initial random phase ϕs1 +ϕs2, their interference term will survive in the ensemble average, leading to the constructive or destructive two-photon interference. Later, Mandel gave a detailed theoretical analysis of HBT-type two-photon interference between two independent sources with random phases, and predicted that the visibility of two-photon interference fringes has a maximum value of 50% for classical light . In general, for the case of two-photon interference with independent light sources, the coincidence counts consist of two parts: (i) The self-correlation part with the pair of photons from the same source which contributes a constant to the correlation function. (ii) The cross-correlation part with the pair of photons from different sources, which is dominated by the superposition of the two indistinguishable paths as depicted in Fig. 1, resulting in the two-photon interference fringes. It is the existence of part (i) that will low down the visibility of two-photon interference fringes. For a quantum source such as the single-photon state, the self-correlation contribution could be eliminated, and therefore giving rise to a 100%-visibility two-photon interference . However, for a classical light, the self-correlation always contributes, which makes the visibility of two-photon interference fringes not exceeding 50%. This property of two-photon interference with independent random-phase sources was further discussed by Paul , Ou and Klyshko , and confirmed experimentally for both the quantum light [10–12] and classical light [13–15]. Nevertheless, when one considers the multi-photon interference, the visibility could be higher than 50% for classical lights based on the third- and higher-order coherence [16–19]. In this paper, without the introduction of third- and higher-order correlation measurement, we explore another way to achieve high visibility two-photon interference with classical light. Instead of reducing or even removing the self-correlation contribution as in the case of quantum sources, we increase the cross-correlation contribution by introducing the superposition of multiple indistinguishable two-photon paths (path number >2) to enhance the two-photon interference effect of a classical light. This could be practically realized by introducing a random-phase grating to modulate the wavefront of a coherent light. The paper is organized as follows. In Sec. 2, we described the structure of the random-phase grating, which was followed by a detailed theoretical study on both the first-order and the second-order spatial correlations of a coherent light transmitting through the random-phase grating. The experimental verification on the theoretical predictions was given in Sec. 3. And finally, we summarized the paper in Sec. 4. 2. Theoretical model and results 2.1. Random-phase grating The random-phase grating is shown schematically in Fig. 2(a). It is a transmission N-slit mask with specially designed random-phase structure shown in the inset of Fig. 2(a), in which b is the transmission slit width and d is the distance between neighboring slits, respectively. The phase encoded on the nth transmission slit of the grating is designed to be Φ(xs, t) = rect((xs − nd)/b)(n − 1)ϕ(t), where rect(xs) is the one-dimensional rectangular function, xs is the position on the grating plane with xs = nd being the center of the nth slit of the grating, n is a positive integer and the elementary phase ϕ(t) is a temporally random phase (In the following, we will use ϕ to represent ϕ(t) for simplicity but without causing confusion). In this way, a random phase (n − 1)ϕ will be encoded on the light wave transmitting through the nth slit. Such a random-phase grating can be realized through a spatial light modulator (SLM) in practice, as we will demonstrate experimentally in Sec. 3. In the following, we will consider the single-photon and two-photon interference effects when a collimated coherent light transmits through the random-phase grating, as shown in Fig. 2(b). 2.2. Theoretical results To clearly illustrate the single-photon and two-photon interference effect of the light field transmitting through the random-phase grating, we will calculate the first-order and second-order correlation functions of the transmitting field in the Fraunhofer zone, i.e., in the focal plane of a lens put behind the random-phase grating, as shown in Fig. 2(b). For simplicity, we assume that the coherent light incident normally onto the random-phase grating is a plane wave and a single-mode one, in one-dimensional case, the field operator on the detection plane is expressed as [20–22]Eq. (1) into Eq. (2) and taking the condition 〈eiϕ〉 = 0, one gets Eq. (1) into Eq. (4) and taking the condition 〈eiϕ〉 = 0, the second-order spatial correlation function can be deduced as (see Appendix A) Fig. 1: (1) one photon transmitting through the mth slit goes to the detector D1, while the other transmitting through the nth slit goes to the detector D2; and (2) one photon transmitting through the mth slit goes to the detector D2, while the other transmitting through the nth slit goes to the detector D1. Here we introduce the delta function δ(m − n) to show that there is only one path when the two photons transmit through the same slit to trigger a coincidence count. It can also be found that a random phase (m + n − 2)ϕ will be encoded on the amplitudes of the twin two-photon paths as represented in the first line of Eq. (5). For a N-slit random-phase grating as shown in Fig. 2, there could be many such twin two-photon paths originated from different pairs of slits (m, n), and the amplitudes of those twin paths with equal (m + n) will contain the same random phase (m + n − 2)ϕ. These twin two-photon paths are indistinguishable in principle. In this way, multiple different but indistinguishable two-photon paths are introduced through the N-slit random-phase grating. As shown in the second line of Eq. (5), the amplitudes of all different but indistinguishable two-photon paths with the same random phase lϕ(l = 0, 1,⋯ , 2N − 2) are superposed to calculate their contributions to the coincidence probability, and then the coincidence probability contributions from those with different random phases lϕ are added to get the total coincidence probability. Next, we will show that such a superposition of multiple two-photon amplitudes would enhance the two-photon interference, leading to high-visibility two-photon interference for classical light. Thus, the normalized second-order spatial correlation function can be calculated as (see Appendix B)22], and therefore can be called as multiple-slit two-photon interference function. It is seen that g(2)(x1, x2) in Eq. (6) is a sum of (2N −1) multiple-slit two-photon interference functions sin2((l′ +1)(β1 −β2)d/2) / sin2((β1 − β2)d/2) introduced by the random-phase grating, each one is associated with a group of different but indistinguishable two-photon paths which are characterized by the same random phase lϕ(l = 0, 1,⋯ , 2N − 2) in Eq. (5). These multiple-slit two-photon interference functions are periodical functions of the position difference (x1 − x2) with the same period Λ = λf/d in the paraxial approximation, which is exactly the same as that of the multiple-slit single-photon interference pattern of a normal grating with respect to the position x on the detection plane . Therefore, two-photon interference fringes can be observed on the detection plane. The visibility of two-photon interference fringes can be calculated through a formula , where and are the peak and valley, respectively, of the interference fringes described by Eq. (6). It is not hard to find out that these multiple-slit two-photon interference functions are peaked at the same position differences satisfying (β1 − β2)d = ±2nπ (n = 0, 1, 2, ⋯) due to the constructive interference effect, i.e., when the phase difference among different but indistinguishable two-photon paths are an integer multiple of 2π. The constructive interference peak for each multiple-slit two-photon interference function is (l′ + 1)2, and therefore, one can get the interference peak of g(2)(x1, x2) to be (2N2 + 1)/(3N), according to Eq. (6). On the other hand, the minimum of g(2)(x1, x2) is achieved at the condition (β1 − β2)d = ±(2n + 1)π (n = 0, 1, 2, ⋯) due to the destructive interference effect among multiple two-photon paths. However, the minimum of g(2)(x1, x2) is not zero but calculated to be 1/N due to the existence of the cases when the two photons transmit through the same slit of the grating. Therefore, the visibility of the two-photon interference fringes is found to be V = (N2 − 1)/(N2 + 2), which grows quickly with the increase of slit number N and exceeds 50% when N > 2, as shown in Fig. 3. In the following Sec. 3, we will give an experimental verification on the high-visibility two-photon interference fringes described by Eq. (6) for a coherent light transmitting through the random-phase gratings. 3. Experimental demonstration and discussions Experimental setup — Figure 4 shows the experimental setup that we used to measure the two-photon interference effect of the light field scattering from the random-phase grating. In our experiments, a single mode, continuous-wave laser with a wavelength of 780 nm was introduced as the light source, which was expanded and collimated through a beam expander to obtain a plane wave. The expanded and collimated light beam was then reflected by a beam splitter BS and incident normally onto a random-phase grating. Here the random-phase grating was composed of a N-slit amplitude mask (b = 72 μm and d = 400 μm) and a reflection-type phase-only SLM (HEO 1080P from HOLOEYE Photonics AG, Germany) put just behind the mask. The light first transmitted through the N-slit amplitude mask, and then was reflected back from the SLM and finally re-transmitted through the N-slit amplitude mask again. Here we put the SLM as close as possible to the mask, ensuring that the light goes in and out of the same slit of the mask. The SLM provided the desired phase structure on the N-slit mask as shown in the inset of Fig. 2(a). At last, the light waves scattered from the random-phase grating were collected by a lens L with a focal length f = 80 cm. Both the intensity and the second-order spatial correlation measurements were performed on the focal plane of the lens L by using a charge coupled device (CCD) camera with a frame acquisition time of 0.79 ms. Figure 5 shows the measured single-photon and two-photon interference patterns on the detection plane (i.e., the focal plane of the lens L) at different conditions, in which the empty circles are the experimental results while the red curves are the theoretical fits, respectively. Results for traditional grating — When there is no electric signal loaded on the SLM, our experimental configuration is essentially the same as a typical setup to measure the single-photon interference of a traditional N-slit grating. In the experiment, we measured the stationary single-photon interference patterns of the N-slit gratings (N = 2, 3, 4 and 5, respectively). The results are shown in the first column of Fig. 5. As expected, stationary single-photon interference fringes described by the multiple-slit single-photon interference function sin2(Nβd/2)/sin2(βd/2) were observed. The period between the neighboring principal intensity peaks was measured to be 1.57 mm on the detection plane, and (N − 2) sub-peaks appear between the two neighboring principal peaks of the stationary single-photon interference fringes. Note that the normalized second-order spatial correlation function g(2)(x1, x2) in this case was confirmed to be a unity (not shown in Fig. 5). Results for random-phase grating — When the SLM was loaded with the random phases, the random-phase grating was constructed. In this case, there should be no stationary single-photon interference fringes since the phase difference between every two slits changes randomly with time. The second column of Fig. 5 shows the experimental results, in which each one is an intensity average over 10000 frames of the intensity distribution measured by the CCD camera, corresponding to 10000 realization of the random elementary phase ϕ which is uniformly distributed within [0, 2π]. Note that the elementary phase ϕ was kept to be a fixed value with a duration time of 500 ms for each intensity measurement, but it changed randomly from one measurement to the other. It is seen that the single-photon interference fringes were almost erased, leaving an intensity distribution enveloped by a diffraction profile (see the respective red curves) described by Eq. (3). One notes that there are still some residual intensity fluctuations which deviate from the intensity envelop predicted by Eq. (3). This is mainly due to the unavoidable phase flicker of the SLM during each intensity measurement. Although the single-photon interference fringes disappear with the random-phase grating, two-photon interference fringes appear as predicted by Eq. (6). The third column of Fig. 5 shows the measured second-order spatial correlation function g(2)(x1, x2), which is calculated through a formula g(2)(x1, x2) = 〈I(x1)I(x2)〉/(〈I(x1)〉〈I(x2)〉) [18,23] by using the same 10000 frames of the measured intensity distributions as those used in the second column of Fig. 5. Here the red curves are the theoretical fits using Eq. (6). It is seen that the second-order correlation function exhibits itself in the form of high quality interference fringes, in good agreement with the theoretical prediction by Eq. (6). It is seen from the experimental results shown in the third column of Fig. 5 that, the two-photon interference fringes are peaked at the position differences x1 − x2 = ±2nπf / (kd) and minimized at the position differences x1 − x2 = ±(2n + 1)πf / (kd) (n = 0, 1, 2, ⋯), respectively. The period of the two-photon interference fringes Λ was measured to be 1.57 mm, in good agreement with the prediction of Eq. (6). On the other hand, sub-peaks typical for the single-photon interference fringes shown in the first column of Fig. 5 were not observed in the two-photon interference fringes in the third column of Fig. 5. This is due to the fact that g(2)(x1, x2) is a sum of (2N − 1) different multiple-slit two-photon interference functions (see Eq. (6)), and these different multiple-slit two-photon interference functions are always in phase at their principal peaks but out of phase at the sub-peaks. Moreover, the visibility of the two-photon interference fringes was measured to be 44.9%, 59.1%, 62.3% and 71.9% for the N-slit random-phase gratings with N = 2, 3, 4 and 5, respectively. As predicted by Eq. (6), the visibility of the two-photon interference fringes increases with the increase of the slit number N of the random-phase gratings and surpasses 50% when N > 2. Further discussions — One may note that, except for the N = 2 case, the random phases encoded on the slits of the random-phase grating are not fully independent but indeed correlated with respect to each other. Therefore, our case is different from the case discussed by Mandel , where classical lights with fully independent random phases are considered, and which in fact corresponds to the N = 2 case in our configuration. For classical lights with fully independent random phases, the visibility of two-photon interference fringes cannot exceed 50%, as also confirmed by the N = 2 case in our configuration (V = 44.9%, see the two-photon interference fringes in the top first one of the third column in Fig. 5). More importantly, our results show that, by appropriately controlling the random phase structure encoded on a coherent light field, one could achieve two-photon interference fringes with the visibility exceeding 50%. It is known that controlling optical phase plays a key role in the single-photon interference effect , our results show that it may also play an important role in controlling the high-order coherence of light. In summary, we have designed a kind of two-photon grating with a special random-phase structure, through which the single-photon interference is smeared out but the two-photon interference appears. With such a random-phase grating, superposition of multiple indistinguishable two-photon paths is introduced, which leads to high-visibility two-photon interference fringes of classical light. Theoretically, the visibility of the two-photon interference fringes for a coherent light transmitting through a N-slit random-phase grating reaches (N2 −1)/(N2 +2). Experimentally, the visibility of the two-photon interference fringes with a N-slit random-phase grating (N = 2, 3, 4 and 5) was measured to be 44.9%, 59.1%, 62.3% and 71.9%, respectively. The results show the possibility to control the high-order coherence of light through optical phase. For the case when a coherent light, which is the eigenstate of the annihilation operator â, is incident normally onto the random-phase grating as shown in Fig. 2(b), one arrives atEq. (1) into Eq. (4). After taking the square of mould in Eq. (7), one needs to do the ensemble average for the terms Eq. (7), one arrives at Eq. (5). This work was supported by the 973 program ( 2013CB328702), the CNKBRSF ( 2011CB922003), the NSFC ( 11174153, 90922030 and 10904077), the 111 project ( B07013), and the Fundamental Research Funds for the Central Universities. References and links 1. P. Dirac, The Principles of Quantum Mechanics, 2nd edition (Oxford University, 1935). 2. R. Brown and R. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177(4497), 27–29 (1956) [CrossRef] . 3. R. Brown and R. Twiss, “A test of new type of stellar interferometer on sirius,” Nature 178(4541), 1046–1048 (1956) [CrossRef] . 4. U. Fano, “Quantum theory of interference effects in the mixing of light from phase-independent sources,” Am. J. Phys. 29(8), 539–545 (1961) [CrossRef] . 5. J. Liu and G. Zhang, “Unified interpretation for second-order subwavelength interference based on Feynmans path-integral theory,” Phys. Rev. A 82(1), 013822 (2010) [CrossRef] . 6. L. Mandel, “Photon interference and correlation effects produced by independent quantum sources,” Phys. Rev. A 28(2), 929–943 (1983) [CrossRef] . 7. H. Paul, “Interference between independent photons,” Rev. Mod. Phys. 58(1), 209–231 (1986) [CrossRef] . 9. D. Klyshko, “Quantum optics: quantum, classical, and metaphysical aspects,” Phys. Usp. 37(11), 1097–1123 (1994) [CrossRef] . 11. E. J. S. Fonseca, C. H. Monken, and S. Pádua, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. 82(14), 2868–2871 (1999) [CrossRef] . 12. K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. 89(21), 213601 (2002) [CrossRef] [PubMed] . 13. G. Scarcelli, A. Valencia, and Y. Shih, “Two-photon interference with thermal light,” Europhys. Lett. 68(5), 618–624 (2004) [CrossRef] . 14. J. Xiong, D. Cao, F. Huang, H. Li, X. Sun, and K. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. 94(17), 173601 (2005) [CrossRef] [PubMed] . 15. Yan-Hua Zhai, Xi-Hao Chen, Da Zhang, and Ling-An Wu, “Two-photon interference with true thermal light,” Phys. Rev. A 72(4),043805 (2005) [CrossRef] . 16. I. Agafonov, M. Chekhova, T. Iskhakov, and A. Penin, “High-visibility multiphoton interference of Hanbury Brown-Twiss type for classical light,” Phys. Rev. A 77(5), 053801 (2008) [CrossRef] . 17. D. Cao, J. Xiong, S. Zhang, L. Lin, L. Gao, and K. Wang, “Enhancing visibility and resolution in Nth-order intensity correlation of thermal light,” Appl. Phys. Lett. 92(20), 201102 (2008) [CrossRef] . 18. X. Chen, I. Agafonov, K. Luo, Q. Liu, R. Xian, M. Chekhova, and L. Wu, “High-visibility, high-order lensless ghost imaging with thermal light,” Opt. Lett. 35(8), 1166–1168 (2010) [CrossRef] [PubMed] . 19. Y. Zhou, J. Simon, J. Liu, and Y. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A 81(4), 043831 (2010) [CrossRef] . 20. R. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130(6), 2529–2539 (1963) [CrossRef] . 21. R. Glauber, “Coherent and incoherent state of radiation field,” Phys. Rev. 131(6), 2766–2788 (1963) [CrossRef] . 22. G. Brooker, Modern Classical Optics (Oxford University, 2003). 23. Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury Brown and Twiss interferometry with interacting photons,” Nature Photonics 4, 721–726 (2010) [CrossRef] .

Start with three pairs of socks. Now mix them up so that no mismatched pair is the same as another mismatched pair. Is there more than one way to do it? Using the statements, can you work out how many of each type of rabbit there are in these pens? This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15! El Crico the cricket has to cross a square patio to get home. He can jump the length of one tile, two tiles and three tiles. Can you find a path that would get El Crico home in three jumps? Chandra, Jane, Terry and Harry ordered their lunches from the sandwich shop. Use the information below to find out who ordered Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families? Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it? My briefcase has a three-number combination lock, but I have forgotten the combination. I remember that there's a 3, a 5 and an 8. How many possible combinations are there to try? There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether! Can you find out in which order the children are standing in this Lorenzie was packing his bag for a school trip. He packed four shirts and three pairs of pants. "I will be able to have a different outfit each day", he said. How many days will Lorenzie be How many different shapes can you make by putting four right- angled isosceles triangles together? Use the information to describe these marbles. What colours must be on marbles that sparkle when rolling but are dark inside? Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread? Imagine that the puzzle pieces of a jigsaw are roughly a rectangular shape and all the same size. How many different puzzle pieces could there be? The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse? Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total. Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag? These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out. These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach. These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out. This challenge is about finding the difference between numbers which have the same tens digit. This challenge focuses on finding the sum and difference of pairs of two-digit numbers. Can you find all the ways to get 15 at the top of this triangle of numbers? This task follows on from Build it Up and takes the ideas into three dimensions! Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens? These activities lend themselves to systematic working in the sense that it helps to have an ordered approach. You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how? This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'. In this maze of hexagons, you start in the centre at 0. The next hexagon must be a multiple of 2 and the next a multiple of 5. What are the possible paths you could take? There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it? Moira is late for school. What is the shortest route she can take from the school gates to the entrance? How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...? My coat has three buttons. How many ways can you find to do up all Winifred Wytsh bought a box each of jelly babies, milk jelly bears, yellow jelly bees and jelly belly beans. In how many different ways could she make a jolly jelly feast with 32 legs? There are 44 people coming to a dinner party. There are 15 square tables that seat 4 people. Find a way to seat the 44 people using all 15 tables, with no empty places. The Red Express Train usually has five red carriages. How many ways can you find to add two blue carriages? Can you arrange 5 different digits (from 0 - 9) in the cross in the You have 5 darts and your target score is 44. How many different ways could you score 44? Cherri, Saxon, Mel and Paul are friends. They are all different ages. Can you find out the age of each friend using the The brown frog and green frog want to swap places without getting wet. They can hop onto a lily pad next to them, or hop over each other. How could they do it? Explore the different snakes that can be made using 5 cubes. In Sam and Jill's garden there are two sorts of ladybirds with 7 spots or 4 spots. What numbers of total spots can you make? These eleven shapes each stand for a different number. Can you use the multiplication sums to work out what they are? Katie had a pack of 20 cards numbered from 1 to 20. She arranged the cards into 6 unequal piles where each pile added to the same total. What was the total and how could this be done? In how many ways could Mrs Beeswax put ten coins into her three puddings so that each pudding ended up with at least two coins? How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green? How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

The topic today in this group theory thread is “sixtors and representations of the Lorentz group”. Consider the group of proper orthochronous Lorentz transformations and the transformation law of the electromagnetic tensor . The components of this antisymmetric tensor can be transformed into a sixtor or and we can easily write how the Lorentz group acts on this 6D vector ignoring the spacetime dependence of the field. Under spatial rotations, transform separately in a well-known way giving you a reducible representation of the rotation subgroup in the Lorent orthochronous group. Remember that rotations are a subgroup of the Lorentz group, and it contains Lorentz boosts in addition to those rotations. In fact, in the space of sixtors and they are thus a reducible representation, a direct sum group representation. That is, rotations leave invariant subspaces formed by and invariant. However, these two subspaces mix up under Lorentz boosts! We have written before how transform under general boosts but we can simplify it without loss of generality and for some matrices . So it really seems that the representation is “irreducible” under the whole group. But it is NOT true! Irreducibility does not hold if we ALLOW for COMPLEX numbers as coefficients for the sixtors/bivectors (so, it is “tricky” and incredible but true: you change the numbers and the reducibility or irreducibility character does change. That is a beautiful connection betweeen number theory and geometry/group theory). It is easy to observe that using the Riemann-Silberstein vector and allowing complex coefficients under Lorent transformations, such that i.e., it transforms totally SEPARATELY from each other () under rotations and the restricted Lorentz group. However, what we do have is that using complex coefficients (complexification) in the representation space, the sixtor decomposes into 2 complex conjugate 3 dimensional representaions. These are irreducible already, so for rotations alone transformations are complex orthogonal since if you write with and . Be aware: here is an imaginary angle. Moreover, transforms as follows from the following equation: Remark: Rotations in 4D are given by a unitary 4-vector such as and the rotation matrix is given by the general formula If you look at this rotation matrix, and you assign with , the above rotations are in fact the same transformations of the electric and magnetic parts of the sixtor! Thus the representation of the general orthochronous Lorentz group is secretly complex-orthogonal for electromagnetic fields (with complex coefficients)! We do know already that are the electromagnetic main invariants. So, complex geometry is a powerful tool too in group theory! :). The real and the imaginary part of this invariant are also invariant. The matrices of 2 subrespresentations formed here belong to the complex orthogonal group . This group is a 3 dimensional from the complex viewpoint but it is 6 dimensional from the real viewpoint. The orthochronous Lorentz group is mapped homomorphically to this group, and since this map has to be real and analytic over the group such that, as Lie groups, . We can also use the complex rotation group in 3D to see that the 2 subrepresentations must be inequivalent. Namely, pick one of them as the definition of the group representation. Then, it is complex analytic and its complex parameter provide any equivalent representation. Moreover, any other subrepresentation is complex conjugated and thus antiholomorphic (in the complex sense) in the complex parameters. Generally, having a complex representation, i.e., a representation in a COMPLEX space or representation given by complex valued matrices, implies that we get a complex conjugated reprentation which can be equivalent to the original one OR NOT. BUT, if share with original representation the property of being reducible, irreducible or decomposable. Abstract linear algebra says that to any representation in complex vector spaces there is always a complex conjugate representation in the complex conjugate vector space . Mathematically, one ca consider representations in vector spaces over various NUMBER FIELDS. When the number field is extended or changed, irreducibility MAY change into reducibility and vice versa. We have seen that the real sixtor representation of the restricted Lorentz group is irreducible BUT it becomes reducible IF it is complexified! However, its defining representation by real 4-vectors remains irreducible under complexification. In Physics, reducibility is usually referred to the field of complex numbers , since it is generally more beautiful (it is algebraically closed for instance) and complex numbers ARE the ground field of representation spaces. Why is this so? There are two main reasons: 1st. Mathematical simplicity. is an algebraically closed filed and its representation theory is simpler than the one over the real numbers. Real representations are obtained by going backwards and “inverting” the complexification procedure. This process is sometimes called “getting the real forms” of the group from the complex representations. 2nd. Quantum Mechanics seems to prefer complex numbers (and Hilbert spaces) over real numbers or any other number field. The importance of is understood from the Maxwell equations as well. In vacuum, without sources or charges, the full Maxwell equations read These equations are Lorentz covariant and reducibility is essential there. It is important to note that implies that we can choose ONLY one of the components of the sixtor, or , or one single component of the sixtor is all that we need. If in the induction law there were a plus sign instead of a minus sign, then both representations could be used simultaneously! Furthermore, Lorentz covariance would be lost! Then, the Maxwell equations in vacuum should satisfy a Schrödinger like equation due to complex linear superposition principle. That is, if and are solutions then a complex solution with complex coefficients should also be a solution. This fact would imply invariance under the so-called duality transformation However, it is not true due to the Nature of Maxwell equations and the (apparent) absence of isolated magnetic charges and currents!

They are also the simplest oscillatory systems. Periodic motion is defined as the motion that repeats itself after fixed intervals of time. Consequently, there is no mean … The to-and-fro motion of a body is called oscillatory motion. Harmonic motion refers to the motion an oscillating mass experiences when the restoring force is proportional to the displacement, but in opposite directions. The complete oscillation: the complete oscillation is the motion of an oscillating body when it passes by a fixed point on its path two successive times in the same direction. Motion which repeats itself after a fixed interval of time is called periodic motion. Oscillatory motion is defined as the to and fro motion of the … Answer: Any motion that repeats itself at regular interval of time is called periodic motion. Around the sum is 365 h. Oscillatory Motion: When a body moves in to and fro motion over and over again about a fixed point, then its motion is called oscillatory motion. the motion of bob of an oscillating simple pendulum is an oscillatory motion as well as periodic motion, while the motion of the earth around the sun is a periodic motion but not … All the oscillatory motions are not periodic such as the revolution of the earth is periodic but not an oscillatory motion. It is also known as periodic motion. "Oscillatory motion" is motion that repeats over and over again after a time T that is called the "period." In simple harmonic motion, the restoring force is directly proportional to the displacement of the mass and acts in the direction opposite to the displacement direction, pulling the particles towards the mean position. Thus, it is a periodic motion. Oscillatory motion is always periodic motion. Some periodic motion examples are: The earth completes one round around the sun in 356-1/4 days and this motion gets repeated after every 356-1/4 days. Oscillatory motion is a repetitive motion between two or more states. Simple harmonic motion is an important topic in the study of mechanics. Search in book: Search Contents. According to Newton’s law, the force acting on the mass m is given by F =-kxn. Examples: Motion of the pendulum of a clock, motion of planets around the Sun, etc. The periodic motion is the type of motion that gets repeated after a regular interval of time. In some cases of periodic motion, the body moves periodically (back and forth, up and down, to and fro, etc. ) We’ve already encountered two examples of oscillatory motion - the rotational motion of Chapter 5, and the mass-on-a-spring system in Section 2.3 (see Figure 1.1.1).The latter is the quintessential oscillator of physics, known as the harmonic oscillator.Recapping briefly, we get its equation of motion … In oscillatory motion, the fixed point or position about which the body oscillates is called equilibrium position. For e.g. Oscillatory motion is a type of periodic motion. 01. Oscillatory Motion • Periodic motion • Spring-mass system • Differential equation of motion • Simple Harmonic Motion (SHM) • Energy of SHM • Pendulum BRAC University, Summer 2020 Periodic Motion • Periodic motion is a motion that regularly returns to a given position after a fixed time interval. The motion of the pendulum of … The time taken for an oscillation to occur is often referred to as the oscillatory period. Let us consider a string fixed tightly between two walls. The period is the duration of one cycle in a repeating … If an objects motion is periodic, then there is a characteristic time: the time … Oscillatory motions are well defined for damped oscillations, simple harmonic oscillations, and for … Simple Harmonic Motion: A Special Periodic Motion. Simple harmonic motion is the to-and-fro motion of body … f 1 T T 1 f 2 f 2 T Simple harmonic motion: if the restoring force is proportional to the distance fromis proportional to the distance from equilibrium, the motion will be of the SHM type. All oscillatory motions are periodic motions. Ans: All oscillatory motions are said to be periodic motion because each oscillation is completed in a fixed interval of time. Oscillatory motion is a motion in which to and fro movement is done about its mean in fixed interval of time or periodically whereas periodic motion is the motion in which motion is repeated periodically, it is not necessary to have to and fro movement in periodic motion . The angular frequency and period do not depend on the amplitude of oscillation. Period and Frequency. PERIODIC MOTION. But every periodic motion need not be an oscillatory motion. meanwhile, an oscillatory motion is a motion in which particle moves to and fro about a fixed point, for example motion of a pendulum. Periodic motion can be any motion after a certain period that the motion of the previous period repeats. Periodic motion, in physics, motion repeated in equal intervals of time.Periodic motion is performed, for example, by a rocking chair, a bouncing ball, a vibrating tuning fork, a swing in motion, the Earth in its orbit around the Sun, and a water wave.In each case the interval of time for a repetition, or cycle, of the motion is … In mechanics and physics, simple harmonic motion is a special type of periodic motion where the restoring force on the moving object is directly proportional to the object's displacement magnitude and acts towards the object's equilibrium position. Circular motion is a periodic motion; however, it is still not oscillatory as the net force on the particle in a circular motion is never zero and is always directed towards the centre. Periodic motion: motion that repeats itself in a defined cycle. about a fixed point or position, these types of motions are known as Oscillatory motion or Vibratory motion. When the string is plucked and released, it executes to-and … Periodic motion is also exhibited in other cases such as the to-and-fro motion of a sewing machine needle and the movement of the piston of an engine. About OpenStax; About This Book It is essential to keep in mind that every oscillatory motion is periodic; however, every periodic motion may not be oscillatory. a periodic motion is a motion in which a particle repeats its motion after a fixed time period. In addition, oscillatory motion is bounded … A to and fro or back and forth motion of a body along the same path, without any change in shape of the body, is called an oscillatory motion. An example of this is a weight bouncing on a spring. Periodic Motion Periodic motion is motion that repeats itself. Examples of periodic motion are motion of hands of the clock, motion of planets around the sun etc. This fixed interval of time is known as time period of the periodic motion. A periodic motion may or may not be oscillatory. An oscillation can be a periodic motion that repeats itself in a regular cycle, such as a sine wave —a wave with perpetual motion as in the side-to-side swing of a pendulum, or the up-and-down motion of a spring with a weight. for example circular motion, pendulum motion are periodic motions. Examples of oscillatory motion are vibrating strings, swinging of the swing etc. Examples to understand oscillatory motion and periodic motion; Define periodic motion; Understand every oscillatory motion is periodic but every periodic motion need not be oscillatory; Difference between oscillation and vibration, Define time period; Units of time period; Define frequency; Represent displacement as mathematical function of time; Derive T =2π / ω; Knosw that any periodic … Here, k is the constant and x denotes the displace… The systems where the restoring force on a body is directly proportional to its displacement, such as the dynamics of the spring-mass system, are described mathematically by the simple harmonic oscillator and the regular periodic motion … The fixed interval of time after which a periodic motion is repeated is called as the period of that periodic motion. Harmonic Oscillator. Oscillatory motion is defined as the to and fro motion of the body about its fixed position. The oscillations of a system in which the net force can be described by Hooke’s law are of special importance, because they are very common. But every periodic motion need not be an oscillatory motion. Oscillatory motions are motions where an equilibrium point exists. The to-and-fro motion of a body is called oscillatory motion. Periodic motions are motions that repeat itself over time. Examples of periodic motion are motion of hands of the clock, motion of planets around the sun etc. For example, the motion of planets around the Sun is always periodic but not oscillatory. An oscillating movement occurs around an equilibrium point or mean value. Introduction to Oscillatory Motion and Waves; 16.1 Hooke’s Law: Stress and Strain Revisited; 16.2 Period and Frequency in Oscillations; 16.3 Simple Harmonic Motion: A Special Periodic Motion; 16.4 The Simple Pendulum; 16.5 Energy and the Simple Harmonic Oscillator; 16.6 Uniform Circular Motion and Simple Harmonic Motion; … The main difference between simple harmonic motion and periodic motion is that periodic motion refers to any type of repeated motion whereas simple harmonic motion (SHM) refers to a specific type of periodic motion where … are examples where the objects motion "approximately" keeps repeating itself. Harmonic motion is periodic and can be represented by a sine wave with constant frequency and amplitude. Oscillatory and periodic motions are very abundant in nature and are, therefore, very important in many systems. For example, a small object oscil-lating at the end of a spring, a swinging pendulum, the earth orbiting the sun, etc. These are periodic and non-periodic motion. The periodic time: the periodic time is the time taken by an oscillating body to make one complete oscillation, The measuring unit of the … Oscillatory motions are a type of periodic motion. This motion is also called "periodic motion" with a "repeat time" T. Examples of periodic motion include (1) a mass on a spring, (2) the simple pendulum, (3) the motion of a planet like the Earth about the Sun, and (4) the … 8.1.1. Preface to College Physics. It results in an oscillation which, if uninhibited by friction or any other dissipation of … Example period motion: Period of earth around its own axis is 24n. An oscillatory motion is always periodic.

34. Valuing Delayed Annuities. Suppose that you will receive annual payments of $10,000 for a period of 10 years. The first payment will be made 4 years from now. If the interest rate is 5 percent, what is the present value of this stream of payments? 36. Amortizing Loan. You take out a 30-year $100,000 mortgage loan with Please check my computations to the following questions on the attached spreadsheet. I know that my answers for # 3 are wrong and that the correct answers are $4,167.62, $ 4,313.71, and $ 5,001.15 but I can't figure out what I'm doing wrong. I don't know if my other answers are correct or not. Question 3 - Future Value and Mu BE2-4 Becky Sherrick's regular hourly wage rate is $14, and she receives an hourly rate of $21 for work in excess of 40 hours. During a January pay period, Becky works 45 hours. Becky's federal income tax withholding is $95, her FICA tax withheld is $53.20, and she has no voluntary deductions. Compute Becky Sherrick's gross e Question: The $40 million lottery payment that you just won actually pays $2 million per year for 20 years. If the discount rate is 8%, and the first payment comes in 1 year, what is the present value of the winnings? What if the first payment comes immediately? Cathy is saving for her retirement by putting $325 each month into an ordinary annuity. If the annuity is expected to pay an annual interest rate of 8.5%, how much will she save for her retirement in 30 years? Please help with the following problem. Andahl Corporation stock, of which you own 500 shares, will pay a $2-per-share dividend one year from today. Two years from now Andahl will close its doors; stockholders will receive liquidating dividends of $17.5375 per share. The required rate on return on Andahl stock is 15 percent. A couple is planning for the education of their two children. They plan to invest the same amount of money at the end of each of the next 16 years. The first contribution will be made at the end of the year and the final contribution will be made at the time the oldest child enters college. The money will be invested in sec You deposited $1,000 in a savings account that pays 8 percent interest, compounded quarterly, planning to use it to finish your last year in college. Eighteen months later, you decide to go to the Rocky Mountains to become a ski instructor rather than continue in school, so you close out your account. How much money will you r Multiple Choice Questions: Stock's beta, R2 for a stock and a portfolio, present value of an ordinary annuity Please see attached. 1) Stock A has a beta = 0.8, while Stock B has a beta = 1.6. Which of the following statements is most correct? a. Stock B's required return is double that of Stock A's. b. An equally weighted portfolio of Stock A and Stock B will have a beta less than 1.2. c. If market participants become more risk averse, the required re Multiple Choice- Annuities ________________________________________ Solution The future value of a lump sum at the end of five years is $1,000. The nominal interest rate is 10 percent and interest is compounded semiannually. Which of the following statements is most correct? d. Both statements b and c are correct. e. The future value of a lump sum at the end of five years is $1,000. The nominal interest rate is 10 percent and interest is compounded semiannually. Which of the following statements is most correct? a. The present value of the $1,000 is greater if interest is compounded monthly rather than semiannually. b. The effective a 1. If you borrow $15,618 and are required to pay the loan back in 7 equal annual installments of $300. What is the interest rate assocated with this loan? 2. Your rich uncle has offered you a choice of one of the three following alternatives. Which one would you take? a) $10,000 now b) $2000 a year for 8 years -equal inv 1. The future value of a $500 ordinary annuity received for three years is $________, assuming an investment rate of 10%. a. 1,655.00 b. 665.50 c. 1,820.50 d. 335.65 2. With an interest rate of 9 percent, your investment would double in about a. 4 years. b. 6 years. c. 8 years. d. 10 years. 3. An ordinary annuity What is the effective return of an investment account that pays a 9.9 APR compounded daily (for 365 days)? Okay let's say the annual interest rate is 10%. a.) What would be the present value of a 5-year ordinary annuity with annual payments of $200? b.)What is the present value if it is a 5-year annuity due (all the rest is 1. Determine the future value of an annuity that pays $5,000 at the end of the next 11 years. Similar securities pay an interest rate of 7%. 2. How much money would you be willing to pay in order to receive $800,000 40 years from today? Assume that your required rate of return on investments is 8% compounded semiannually. 1. Today you borrow $80,000 to finance the purchase of your new sports car. Interest will be 5% compounded monthly. Payments will be made at the beginning of the month. You will repay the loan over 4 years. How much will the payments be? 2 If you borrow $100 and pay back $3600 in 5 years, what annual interest rate are you pay You plan to retire in twenty years. When you retire, you will need $150,000 per year for thirty years with the first payment needed at t=21. You expect to receive $50,000 from a trust at t=12 which you will deposit in your retirement account. At t=10, you plan to take a world cruise that will cost you $15,000 to be paid out o You plan to take a long trip through Europe, leaving in 5 years. You're plan is to save money for the next five years, leave at the end of the fifth year, and then survive on your savings for 3 years. You estimate you can survive in Europe on $10,000 a year. You estimate that your investment account will earn 8% forever. You Your client plans to contribute an equal amount of money each year until her retirement. Her first contribution will come in exactly 1 year; her 10th and final contribution will come in 10 years (on her 85th birthday). How much should she contribute each year to meet her objectives? Your client just turned 75 years old and plans on retiring in exactly 10 years (on her 85th birthday). She is saving money today for her retirement and is establishing a retirement account with your office. She would like to withdraw money from her retirement account on her birthday each year until her death. She would ideall 1. Which of the following should be used to calculate the amount of the equal periodic payments that could be equivalent to an outlay of $3000 at the time of the last payment? a) Amount of 1 b) Amount of an annuity of 1 c) Present value of an annuity of 1 d) Present value of 1 (Please give reason for answer) 1. Assume the current (4) year cost to attend Park University for tuition and books is $12,500. It is estimated that these costs will grow at a 7% annual rate. 2. The money that you annual set aside to meet this financial obligation is expected to earn an estimated 5% annually for the 7-year period (period from age 10 t Assume you now have a child and you are planning for her college education. You would like to make monthly deposits over the next 21 years (first payment to be made one month from today) with the final payments to be made at her 21st birthday(a total of 252 deposits) so that you will be able to cover her expected expenses while 12. Your baby girl, Jessica, was born yesterday!! You have made a decision that you need to start a savings program to fund that future college education. After speaking with members of your finance class you decide to save $150 a month for the next 18 years. You feel you can get 8% average return on the savings over the 18 year Discussion Question 1: Many people, as evidenced by the large payoffs provided for picking 6 out of 53 (or more) numbers, play the lottery. The big choice the winners face: taking a lump sum payment today or an annual payment over 20 years. Is a dollar today worth more than a dollar tomorrow? Why or why not? Which do you prefer 1) Terry Austin is 30 years old and is saving for her retirement. She is planning on making 36 contributions to her retirement account at the beginning of each of the next 36 years. The first contribution will be made today (t = 0) and the final contribution will be made 35 years from today (t = 35). The retirement account will BE2-27 Kilarny Company is considering investing in an annuity contract that will return $20,000 annually at the end of each year for 15 years. What amount should Kilarny Company pay for this investment if it earns a 6% return? Unless stated otherwise, interest is compounded annually and payments are at the end of the year. Explanations should be brief (1 or 2 sentences). 1. Jana, who just turned 55, would like to have an annual annuity of $25,000 paid each year for 15 years, the first payment occurring on her 66th birthday. How much must Jana sa 1. Compounding frequency and future value You plan to invest $2,000 in an individual retirement account (IRA) today at a nominal rate of 8 percent, which is expected to apply to all future years. a. How much will you have in the account after 10 years if the interest is compounded: 1. Annually 2. Semi-Annually 3. Daily What will the monthly payment be if you take out a $100,000 15 year mortgage at an interest rate of 1% per month? In May 1992, a 60 yr old nurse gambled $12 in a Reno casino and walked away with the biggest jackpot in history - $9.3 million. In reality, the jackpot wasn't really worth $9.3 million. The sum was to be paid in 20 annual installments of $465,000 each. What is the present value of the jackpot?

Nonlinear speed controller supported by direct torque control algorithm and space vector modulation for induction motors in electrical vehicles. Direct torque control algorithms for induction motors use reference torque values. Therefore, the transformation of reference speed values into reference torque values is needed. Classical feedback controllers use speed error values for the transformation. In addition, different system parameters are incorporated into the transformation for a better control operation , . As it is well-known, resolution is important for controller calculations and frequency is important at time sharing switching operations , . In this study, starting from the design of Lyapunov function-based nonlinear controller, mechanical speed value and speed error value are used for the control transformation. To obtain the required torque value, reference speed value is transformed by either classical speed control or nonlinear speed control methods. In this study, PI controller is preferred for the classical speed control and reference controller is obtained by using well-known Ziegler-Nichols tuning rule. Unit step function is used for input reference at design operation . For the nonlinear speed control, Lyapunov function-based nonlinear controller is proposed. The design of the proposed controller is based on an energy function and opposite signed dynamics of this function. Zero is fixity point for the functions , . In addition, a parameter of the controlled system is used in the energy functions and the dynamic functions. Pause point is expected to be almost 1 for the control parameters. The torque controller uses a reference value provided to the controller. A gas pedal is used to set a reference speed value. The speed controller transforms the speed value into the reference torque value. The torque controller calculates required voltage vector for the motor and sends the vector to inverter controller. For the motor system, a power electronic inverter is used to invert direct currents to alternative currents. Time sharing-based vector controller sets the required voltage vector. Angular and amplitude values of the vector are obtained by the inverter control system. For experimental evaluations in this study, a rotational loaded system was used. In the design steps, the variation of electromagnetic torque parameter was taken into consideration. Required reference torque was created by using the reference speed value for direct torque controller (DTC). II. DIRECT TORQUE CONTROL Induction machine model is described in the general reference frame by using the following equations: [[bar.V].sup.g.sub.s] = [R.sub.s] x [[bar.i].sup.g.sub.s] + d[[bar.[psi]].sup.g.sub.s]/dt + j x [[omega].sup.g] x [[bar.[psi]].sup.g.sub.s], (1) 0 = [R.sub.r] x [[bar.i].sup.g.sub.r] + d[[bar.[psi]].sup.g.sub.r]/dt + j([[omega].sub.g] - [[omega].sub.r])[[bar.[psi]].sup.g.sub.r], (2) [[bar.[psi]].sup.g.sub.s] = [L.sub.s] x [[bar.i].sup.g.sub.s] + [L.sub.m] x [[bar.i].sup.g.sub.r], (3) [[bar.[psi]].sup.g.sub.r] = [L.sub.r] x [[bar.i].sup.g.sub.r] + [L.sub.m] x [[bar.i].sup.g.sub.s], (4) where [[bar.V].sub.s] is supply voltage signal of induction motor stator, [[bar.i].sub.s] and [[bar.i].sub.r] are stator and rotor currents, [[omega].sub.g] is general reference speed, [[omega].sub.r] is rotor angular velocity speed, [R.sub.s] and [R.sub.r] are stator and rotor resistances, [[bar.[psi]].sup.g.sub.s] and [[bar.[psi]].sup.g.sub.r] are stator and rotor field linkages. [L.sub.s] represents the parameter of stator leakage inductance, [L.sub.r] represents the parameter of rotor leakage inductance, and [L.sub.m] represents the parameter of mutual inductance. The superscript "g" refers to reference frame , . Mechanical dynamics of the induction motor is described as follows: [T.sub.e] = [3/2] x [p/2] x Im([[bar.[psi]].sup.g.sub.s] x [[bar.i].sup.g.sub.s]), (5) J x [d[[omega].sub.n]/dt] = J x [2/p] x [d[[omega].sub.r]/dt] = [T.sub.e] - [T.sub.load], (6) where [T.sub.e] is the motor torque, p is the number of motor poles, J is system inertia, [[omega].sub.m] is rotor mechanical angular velocity speed, and Thad is [T.sub.load] torque value . One of the six non-zero and two zero voltage vectors is selected by the torque controller and required switch positions are sent to the inverter , : [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8) where [DELTA][T.sub.e] and [DELTA][[bar.[psi]].sup.g.sub.s] are the symbols of torque and flux errors, respectively. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] represent the reference values of [T.sub.e] and [[bar.[psi]].sup.g.sub.s], respectively. For selecting voltage vector, the errors must be taken into the hysteresis bands. The [[bar.V].sup.g.sub.s] vectors can be calculated using (9) [[bar.V].sup.g.sub.s]] = 2/3 x [V.sub.dc] x ([S.sub.a] + [e.sup.jx2.[pi]/e] x [S.sub.b] + [e.sup.jx4.[pi]/3] x [S.sub.c]), (9) where [S.sub.a], [S.sub.b] and [S.sub.c] represent stator phase A, phase B, and phase C, respectively. When the vector is selected, stator flux is rotated to desired frequency, which is inside a specified band. If the motor stator ohmic drops are neglected, stator flux dynamic is calculated using (10) d[[bar.[psi]].sup.g.sub.s] = [[bar.V].sup.g.sub.s] x [DELTA]T. (10) If (11) below and (5) are used together, a sinusoidal function of [gamma] is obtained on motor torque. [gamma] is the angle value between stator and rotor fields of induction motor. The angle is adapted and required stator field is obtained using (13) and (14), respectively : [[bar.i].sub.s] = [[bar.[psi]].sup.g.sub.s]/[sigma][L.sub.s] - [[L.sub.m]/[sigma][L.sub.s][L.sub.r]]/[[bar.[psi]].sup.g'.sub.r] and [sigma] = 1 - [L.sup.2.sub.m]/[L.sub.s][L.sub.r], (11) [T.sub.e] = [3/2][p/2][[L.sub.m]/[sigma] x [L.sub.s] x [L.sub.r]] x [absolute value of [[bar.[psi]].sup.g.sub.s]] x [absolute value of [[bar.[psi]].sup.g'.sub.s]] x sin [gamma], (12) [[bar.[psi]].sup.s.sub.s] = [integral]([[bar.V].sup.s.sub.s] - [[bar.i].sup.s.sub.s] x [R.sub.s])dt, (13) [T.sub.e] = 3/2 x p x ([[phi].sup.s.sub.ds] x [i.sup.s.sub.qs] - [[phi].sup.s.sub.qs] x [i.sup.s.sub.ds]), (14) where the superscript "s" refers to stator reference frame. [[phi].sup.s.sub.ds] is the real axis component of stator magnetic field space vector fixed coordinate system, [[phi].sup.s.sub.qs] is the imaginary axis component of stator magnetic field space vector fixed coordinate system, [i.sub.ds] is the real axis of stator flux in the fixed coordinate system, and [i.sub.qs] is the imaginary axis of stator flux in the fixed coordinate system. Measured stator voltage and stator current signals are represented with [[bar.V].sup.s.sub.s] and [[bar.i].sup.s.sub.s]. [[bar.[psi]].sup.s.sub.s] represents calculated stator field vector. [DELTA][[bar.[psi]].sup.g.sub.s] and [DELTA][T.sub.e] states are enough for voltage vector selection . The details of the proposed control system and data flow in the proposed control system are shown in Fig. 1. III. TIME SHARING-BASED SPACE VECTOR MODULATION The disadvantage of classical direct torque control systems is the use of inverter control modulator. This disadvantage is reinforced with time sharing-based space vector modulator which supplies required voltage vector at angle position. The hysteresis band values and flux vector components are placed into the time sharing-based space vector modulator. Measured voltage and current parameters are used for the calculation of stator flux vector. These parameters are also used for the calculation of electromagnetic torque. Torque error value and flux vector positions on their hysteresis bands are enough for selecting required supply voltage vector , . The position, stator flux band position and torque error band position are used in three different cases. In the first case, flux angular position [[epsilon].sub.R]: 0[degrees] ... 60[degrees] is placed in the band area and both of the band values are positive. First active vector is selected as V1 and second active vector is selected as [[bar.V].sub.2]. [t.sub.vectornumber] is relevant vector switching time and [t.sub.total]. is total switching period. Then, [[epsilon].sub.R] is determined in 0[degrees] ... 360[degrees] circular area. When these parameters are used, following time parts are obtained: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15) In the second case, [[epsilon].sub.R]: 120[degrees] ... 180[degrees] is placed in the band area, the flux band value is positive and torque error band value is equal to zero. This means [t.sub.1] = [t.sub.total] and [t.sub.2] = [t.sub.0] = [t.sub.7] = 0 time values must be chosen in time domain. In the third case, torque error band value and stator flux band value have negative signs and [[epsilon].sub.R]: 0[degrees] ... 60[degrees] angular position. Then, [[bar.V].sub.5] and [[bar.V].sub.6] active vectors are selected and switching time parts are calculated as follows [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16) IV. PERFORMANCE EVALUATIONS In the proposed system, nonlinear speed controller and time sharing-based space vector modulator are integrated into the induction motor control system. For experimental evaluations in this study, induction motor parameters, [R.sub.1] = 1.027 [OMEGA], [R.sub.2] = 1.475 [OMEGA], [L.sub.0] =112.7 mH, [L.sub.[sigma]1] = [L.sub.[sigma]2] = 8.07 mH and J = 0.089 kg.[m.sup.2]. DC supply voltage is 336 V. To compare the proposed system with pulse width controlled system, time constant, 20 [micro]s, and carrier frequency, 20 kHz, are used for discrete time applications with pulse width modulator controlled inverter in MATLAB. Sampling time, 50 [micro]s, and [t.sub.total] = 800 [micro]s are used for [down arrow]discrete time application system with time sharing-based space vector modulator controlled inverter in MATLAB. Fig. 2 shows the values of the block parameters of insulated gate bipolar transistor (IGBT) inverter used in this study. Due to the disadvantage that the PCI card added to the system together with the optoelectronic switches has only one analog to digital converter and one digital to analog converter, stator currents-voltages and encoder signals slide on time phase. Hence, in this system, desired performance characteristics may not be obtained by control algorithms. Upper valued time parameters are used for pulse width modulator-based system. DC supply voltage is selected as 141 V for both systems. If voltage signals are taken into consideration, the result of the pulse modulator is 110-155 V and the result of the space modulator is 140 [+ or -] 8 V. Desired sinusoidal current signal, [i.sub.continuous] = 2-sin(2[pi] x 50 x t)A, is generated in the stator windings by the space vector modulator controlled inverter. As it is well-known, circular shaped flux is desired for the best performance of induction motors , . Fig. 3 (a) and (b) show the results of stator phase-A voltage signals obtained from the pulse width modulator and the time shared space vector modulator, respectively. Fig. 4 (a) and (b) show the results of stator phase-A current signals obtained from the pulse width modulator and the time shared space vector modulator, respectively. Fig. 5 (a) and (b) show the results of stator flux positions in D-Q axis obtained from the pulse width modulator and the time shared space vector modulator, respectively. It is seen that desired flux structure is formed by the space vector modulator. The reason of axis slidings is the PCI card used for analog to digital and digital to analog conversions. In Fig. 6 (a) and (b), the positions of rotor mechanical speeds in time domain obtained from the pulse width modulator and the time shared space vector modulator are shown. After 0.1 sec, pulse width controlled system takes its continuous time position. On the other hand, the proposed space vector controller is slower. It takes its continuous time position after 0.4 sec. But, ~[+ or -] 80rpm fluctuation of the pulse width controlled motor shows that its performance is worse in continuous time. To compare the proposed system with classical feedback control and Lyapunov function, 10 (is time constant is used for all discrete time simulations in MATLAB. The same system parameters used in the previous experiments are used. As shown in Fig. 7, stator phase-A currents have nearly same characteristics for both the systems. Also, as shown in Fig. 8, stator flux positions in D-Q axis have nearly As shown in Fig. 9 (a) and (b), the PI controller is 25% faster than the proposed nonlinear controller. However, different from the PI controller, overshoot value of the proposed controller is zero. In Fig. 10 (a) and (b), stator electromagnetic torque positions in time domain obtained from the PI controller and the proposed nonlinear controller are shown. In this paper, the details of nonlinear speed controller supported by direct torque control algorithm and space vector modulation are explained. With the results of comparative experimental studies, the performance of the proposed system and its advantages over classical controllers are shown. Proposed nonlinear speed controller has several advantages over classical controllers such as the use of different Lyapunov functions and easy integration of different system parameters. Manuscript received June 24, 2012; accepted April 19, 2013. O. Ellabban, J. V. Mierlo, and P. Lataire, "A Comparative Study of Different Control Techniques for an Induction Motor Fed by a Z-Source Inverter for Electric Vehicles", in Proc. of International Conference on Power Engineering, Energy and Electrical Drives, Malaga, Spain, 2011, pp. 1-7. H. Ouadi, F. Giri, A. Elfadili, L. Dugard, "Induction machine speed control with flux optimization", Control Engineering Practice, vol. 18, pp. 55-66, 2010. [Online]. Available: http://dx.doi.org/10.1016/ j.conengprac.2009.08.006 H. Miranda, P. Cortes, J. I. Yuz, J. Rodriguez, "Predictive Torque Control of Induction Machines Based on State-Space Models", IEEE Transactions on Industrial Electronics, vol. 56, no. 6, pp. 19161924, 2009. [Online]. Available: http://dx.doi.org/10.1109/ TIE.2009.2014904 M. Pucci, "Direct field oriented control of linear induction motors", Electric Power Systems Research, vol. 89, pp. 1-7, 2011. K. Ogata, Modern Control Engineering. New Jersey: Prentice Hall, 2002, pp. 53-79. J. E. Slotine, W. Li, Applied Non-Linear Control. New Jersey: Prentice Hall, 1991 pp. 383-453. H. K. Khalil, Nonlinear Systems. New Jersey: Pearson Education, 2000, pp. 87-181. N. R. N. Idris, A. H. M. Yatim, "Direct Torque Control of Induction Machines with Constant Switching Frequency and Reduced Torque Ripple", IEEE Transactions on Industrial Electronics, vol. 51, no. 4, pp. 758-767, 2004. [Online]. Available: http://dx.doi.org/ 10.1109/TIE.2004.831718 P. Vas, Sensorless Vector and Direct Torque Control. Oxford: Oxford Science Publications, 1998, pp. 263-352. H. F. A. Wahab, H. Sanusi, "Simulink Model of Direct Torque Control of Induction Machine", American Journal of Applied Sciences, vol. 5, pp. 1083-1090, 2008. [Online]. Available: http://dx.doi.org/10.3844/ajassp.2008.1083.1090 K. Gulez, A. A. Adam, H. Pastaci, "Passive Filter Topology to Minimize Torque Ripples and Harmonic Noises in IPMSM Derived with HDTC", International Journal of Electronics, vol. 94, no. 1, pp. 23-33, 2007. [Online]. Available: http://dx.doi.org/ 10.1080/00207210601070911 C. M. Ong, Dynamic Simulation of Electric Machinery Using MATLAB/Simulink. New Jersey: Prentice Hall, 1998, pp. 415-438. D. Casadei, G. Serre, and K. Tani, "Implementation of a Direct Control Algorithm for Induction Motors Based on Discrete Space Vector Modulation", IEEE Transactions on Power Electronics, vol. 15, no. 4, pp. 769-777, 2000. [Online]. Available: http://dx.doi.org/10.1109/63.849048 K. B. Lee and F. Blaabjerg, "Sensorless DTC-SVM for Induction Motor Driven by a Matrix Converter Using a Parameters Estimation Strategy", IEEE Transactions on Industrial Electronics, vol. 57, no. 3, pp. 512-521, 2008. [Online]. Available: http://dx.doi.org/ 10.1109/TIE.2007.911940 I. Aliskan (1), K. Gulez (2), G. Tuna (3), T. V. Mumcu (2), Y. Altun (2) (1) Department of Electrical and Electronics Engineering, Bulent Ecevit University, Zonguldak, Turkey (2) Department of Control and Automation Engineering, Yildiz Technical University, 34220, Istanbul, Turkey (3) Department of Computer Programming, Edirne Vocational School of Technical Sciences, Trakya University, Edirne, Turkey |Printer friendly Cite/link Email Feedback| |Author:||Aliskan, I.; Gulez, K.; Tuna, G.; Mumcu, T.V.; Altun, Y.| |Publication:||Elektronika ir Elektrotechnika| |Date:||Jun 1, 2013| |Previous Article:||On-line partial discharge detection for high voltage rotating machines using inductive method.| |Next Article:||Modelling of Neuro Fuzzy controller for negative output KY boost converter voltage ripple reduction.|

New theory solves uncertainty principle in quantum mechanics. The uncertainty will continue forever since the scientists assume the space of subatomic particles is the same space that we and large object exists in. The subatomic particles space is different it has extra dimensions which we cannot feel or recognize. The reason why we see this strange behavior of subatomic particles because we think in three-dimensions world and assume that these subatomic particles as well move in this three-dimensions world. Lets assume the subatomic particles have more dimensions via which to move through. One or two or three extra dimensions. Or more. When the scientists and mathematicians manage to formulate the specification of these subatomic dimensions and formulate their relation with our three-dimensions then the certainty will prevail and the Uncertainty principle will vanish. Some times we notice strange behaviors of subatomic particles as if they pop in then pop out from somewhere else. We see it strange because we think in three-dimensions world. The truth is very simple and very normal, all what is happening, the subatomic particles are moving very normal in their dimensions so when they take the forth dimension as a root (where time does not function there. Or this dimension has its own time different from our time), it looks to us as if they are poping in from one place then poping out from different place with no time at all. When you make the formulas and it does not work, then you must take this in consecration. The magnetic force either can not function in other dimensions, or it is deformed so it behaves differently in other dimensions. The same applies to other forces like electric force. You must not take what functions in our dimensions and apply it in other dimensions. Time and force change in other dimensions. Even particles like the electron changes, for example the electron is a particle in some dimensions and change to waves in other dimensions. That is why. Since we live in three-dimensions world and in their specification where they extend forever for billions of light years. Therefore our mind set can only imagine any other dimension to have the same specification of having infinite distance like our dimensions. We must know that the subatomic dimensions have different specification than our dimensions. These subatomic dimensions they are huge and colossal for the subatomic particles. May be the ratio of the size of these subatomic dimensions to the size of the subatomic particles, is the same ratio of human size to our universe. So the subatomic universe must have different values of time and different interaction with forces. The universe of subatomic particles will have totally different behavior than ours where only great minds can image it and great mathematicians can find the formulas by which it function. The ratio of one meter to the diameter of our universe = 8.61 E26 Plank constant = 1.986 E25 J.m So they are so similar. From this we can assume that the subatomic universe has some relation with Plank constant. We should consider this world consists from three universes one is smaller than the other in a ratio of plank constant. Our universe, it is the universe in the middle. Dark matter universe, it is so huge. If we divide our universe diameter by dark matter universe diameter we will get plank constant. 6.62607004 × 10-34 Subatomic universe, it is so tiny. If we divide Subatomic universe diameter by our universe diameter we will get plank constant. 6.62607004 × 10-34 All these three universes coincide in each other. Each universe has three dimensions. But each universe has a limit after that its dimension will shop at and does not function any more. For example the subatomic universe its dimensions are so small as small as plank constant can be. But even though this subatomic universe is every where, if we move all over our universe from one galaxy to another the subatomic universe is in every part of any galaxy and in the space between galaxies. But the subatomic universe only has effect and function within the very small limit of its dimensions. So when a particle moves inside subatomic universe it will behave totally different in this subatomic universe, and when it moves out of it to our universe it will behave according to our universe specification. So we live in a 9 dimensions world where three universe co exist together as if one inside the other but that is not the case. Each universe has properties different from other universes. And the mater inside it behaves in a way it seems to us so strange. That is why we find it difficult to understand the matter in the Dark matter universe and we find it difficult to understand the behavior of matter in Subatomic universe. By this new theory we will end the Uncertainty principle once and for all. But believe me, we will find ourselves in a more puzzling problems to solve. We must enjoy these new problems because if we solve them all then we will have dull world with no mysteries to solve and enjoy solving them. In quantum mechanics, the uncertainty principle, also known as Heisenberg's uncertainty principle, is any of a variety of mathematical inequalitiesasserting a fundamental limit to the precision with which certain pairs of physical properties of a particle, known as complementary variables, such as position x and momentum p, can be known.

Porosity and Permeability Calculator This porosity and permeability calculator uses Darcy's law to calculate these two properties of a porous material with a fluid flowing through it. The main application for this calculation is in the earth sciences to understand how water, oil, gas, etc., travel through the different layers of the earth. The diagram above represents how we model a fluid moving through a porous substance due to a pressure gradient on either side of it. In this accompanying article to our porosity and permeability calculator, we will: - Present Darcy's law equation; - Explain how to calculate permeability; and - Show how to calculate porosity. You might also be interested in another topic related to fluids, for example, the hydraulic conductivity calculator. Be sure to check it! Darcy's law – how to calculate permeability Darcy's law is an approximation used extensively in the earth sciences to determine the characteristics of a material. It models the flow of a fluid through a porous medium. The equation for Darcy's law is: - – Discharge rate, in units of volume per time: meet this quantity at the coefficient of discharge calculator; - – Permeability of the material; - – Cross-sectional area of the material; - – Dynamic viscosity of the fluid (if you know the kinematic viscosity, use the Poise-Stokes converter for an easy fix); - – Distance the fluid travels through the material; and - – Pressure difference either side of the material. We can rearrange this equation to find the permeability in terms of the other quantities that we can measure from experiments: The widely used unit for permeability is appropriately the darcy () or millidarcy (). The dimension of the darcy unit is (length squared). Therefore, you can also represent permeability using the SI units system as (meters squared) – 1 darcy is . What is porosity? How to calculate porosity When we ask "What is the porosity of a substance?", it means to ask how much of the volume of the substance is empty compared to the space taken up by the solid material. The greater the porosity, the more open space per unit volume of material. When using the Darcy equation, the porosity of a substance means how well it allows a fluid to flow through it. The higher the porosity, the better fluids flow through it, but only if the voids are well connected. Let's first look at the equation for fluid flow porosity, and then we'll unpack it and interpret it. The equation for porosity is: - – Porosity of the material; and - – Time taken for the fluid to travel the distance through the substance with cross-sectional area . This equation tells us that the more porous a material is, the quicker the discharge rate is for a given volume of material. Porosity is a dimensionless quantity. How to use the porosity and permeability calculator Using our porosity and permeability calculator is relatively straightforward. First, we define the fluid pressure gradient across the substance. You can either enter the pressure on both sides or input the pressure difference directly. Continue to enter the following terms: - The distance the fluid flows through the material. - The cross-sectional area of the material. - The discharge rate of fluid leaving the material. - The viscosity of the fluid. You will then see that we have enough information to calculate the permeability, and you will be rewarded with a result. To calculate the porosity, input the residence time, which is the time taken for the fluid to move through the material. And there you have it – the porosity of your substance. 💡 If you have values in different units, first change the unit of the variable by clicking on the unit to open a drop-down list of alternative units. What is k in Darcy's law? The term k in Darcy's law represents the permeability of a material and is a measure of how easily a fluid (liquid or gas) can flow through a porous substance, such as sand, rock, etc. How has Darcy's law been verified in nature? Henry Darcy based the law named after him on experiments he performed that involved water flow through beds of sand. His work was the foundation of hydrogeology, one of the earth sciences. How do I calculate coefficient of permeability? To calculate the permeability of a porous material, use Darcy's law equation: - Multiply together the fluid discharge rate, dynamic viscosity, and distance traveled. - Divide the result from Step one by the cross-sectional area of the material multiplied by the pressure difference on either side of the material. - The result is the material's permeability the fluid travels through. What is Darcy velocity? Darcy velocity is defined as the rate flow of a fluid per unit of the cross-sectional area of a porous material. It depends on the porosity of the material and the pressure difference that drives the fluid flow. What is the permeability of soil? 1 to 10 darcy, depending on soil type. A very sandy soil will be more permeable (~10 darcy), whereas a very peaty soil with lots of organic material will be less permeable (~1 darcy). What is the difference between porosity and permeability? Porosity is a measure of the amount of void space in a material. Permeability is a measure of how well fluids flow through a material. You can have a material with very high porosity, but if the voids in the material are not connected, fluids will not flow through it, and its permeability will be lower. What is the porosity of soil? The porosity of soil depends on the soil type. For sandy soils, it is between 0.36 and 0.43, whereas for soil with a lot of clay content, it is between 0.51 and 0.58.

Euclidean & Non-Euclidean Geometries: Development and History by Marvin Jay Greenberg35-year-old-Rick-from-January-2018: Well, I just finished reading a book about the history and development of Non-Euclidean Geometry. 15-year-old-Rick-from-January 1998: Wait, are you me from the future? How did you get here? 35yo-Rick: It would take too long to explain. Just ask Gödel. 15yo-Rick: Okay, but why did you just read a book about geometry? Surely Im still not in school 20 years from now! 35yo-Rick: I read it for fun. 15yo-Rick: Fun?! You think Geometry is fun? Oh no. Please tell me this isnt who I grow up to become. Okay, this sounds crazy to my 15 year old self, and probably crazy to a lot of other people, but I have found that some of the most calming things to read are math books. Something about the order and elegance of a good proof, something I also appreciate in formal logic. I picked this book to read particularly because one of the classes I teach to high schoolers covers Euclids definitions, common notions, and postulates. Its not a math class, but we quickly cover Euclid for philosophical purposes. Mainly we talk about his axiomatic method and how it informed Descartes later on. However, even though its only 2 days of class, I wanted to have a better understanding of non-Euclidean geometry and of the problems with Euclids 5th postulate. Enter Marvin Greenbergs excellent book. There are 10 chapters and 2 appendices that intend to take the reader on a journey through the history of geometry while rigorously inculcating the principles of geometric proofs. Its kind of an all-in-one program, and Greenberg offers ideas on how to teach the book for various classes in the introduction. There are chapters to work through with a math class of moderate skill (Chapters 1-6 and the beginning of 7 [minus all the major exercises]). There are chapters to work through with a class of liberal arts students (Chapters 1, 2, 5, parts of 6 and 7, and 8). There are chapters to work through with a math class of advanced students (Chapters 1-7 with all exercises). Being a glutton for punishment, I decided to work through all the chapters and do the review exercises (but not the major exercises, because Im not that crazy). I found that I was able to follow the discussion well through the first 6 chapters, and I made it part of the way through chapter 7 before I was completely over my head. Chapter 8 was a philosophical overview of the implications of non-Euclidean geometry for philosophy of mathematics; that was a great chapter. Chapters 9 and 10 made my brain hurt, and would have required far more time than I wanted to spend in order to fully grasp. I dont know how much information Ill retain, but there were some great quotes and I think Ive got a good, basic grasp of how non-Euclidean geometry was discovered and what its all about. (Hint: It has nothing to do with Cthulhu. Thanks for confusing me, H. P. Lovecraft!) Another thing this book reinforced for me was my long-held belief that math is the closest thing to actual magic that exists in the world. If there were really a school for witchcraft and wizardry, it would look a lot less like Hogwarts and a lot more like a school full of people doing abstract mathematics and pure analytic geometry. ISBN 13: 9780716799481 This is the definitive presentation of the history, development and philosophical significance of non-Euclidean geometry as well as of the rigorous foundations for it and for elementary Euclidean geometry, essentially according to Hilbert. Appropriate for liberal arts students, prospective high school teachers, math. The first eight chapters are mostly accessible to any educated reader; the last two chapters and the two appendices contain more advanced material, such as the classification of motions, hyperbolic trigonometry, hyperbolic constructions, classification of Hilbert planes and an introduction to Riemannian geometry. Convert currency. Add to Basket. Goodreads helps you keep track of books you want to read. Want to Read saving…. Want to Read Currently Reading Read. Other editions. Enlarge cover. Error rating book. Freeman and Company , 41 Madison Ave. For much of the last half of the twentieth century, college level mathematics textbooks, particularly calculus texts, have included short, marginal, historical blurbs; a short bio of Brook Taylor in the section on Taylor series, for example. Such inclusions can be interesting for the faculty member who has not had much exposure to the history of mathematics or the student with a pre-existing interest. As a student I found these excerpts tantalizing and they surely whetted my appetite for mathematics history. However, as a professor I have found them frustrating as they rarely say enough about the mathematics itself. This is the definitive presentation of the history, development and philosophical significance of non-Euclidean geometry as well as of the rigorous foundations for it and for elementary Euclidean geometry, essentially according to Hilbert. Appropriate for liberal arts students, prospective high school teachers, math. The first eight chapters are mostly accessible to any educated reader; the last two chapters and the two appendices contain more advanced material, such as the classification of motions, hyperbolic trigonometry, hyperbolic constructions, classification of Hilbert planes and an introduction to Riemannian geometry. Marvin J. He received his undergraduate degree from Columbia University, where he was a Ford Scholar.

1 We have studied a number of counting principles and techniques since the beginning of the course and when we tackle a counting problem, we may have to use one or a combination of these principles. The counting principles we have studied are: Inclusion-exclusion principle: n(a B) = n(a) + n(b) n(a B). Complement Rule n(a ) = n(u) n(a). Multiplication principle: If I can break a task into r steps, with m1 ways of performing step 1, m2 ways of performing step 2 (no matter what I do in step 1),..., mr ways of performing step r (no matter what I do in the previous steps), then the number of ways I can complete the task is m1 m2 mr. (This also applies if step i of task amounts to selecting from set Ai with mi elements.) Addition principle: If I must choose exactly one activity to complete a task from among the (disjoint) activities A1, A2,..., Ar and I can perform activity 1 in m1 ways, activity 2 in m2 ways,..., activity r in mr ways, then I can complete the task in m1 + m2 + + mr ways. (This also applies if task amounts to selecting one item from r disjoint sets A1, A2,..., Ar with m1, m2,..., mr items respectively.) Permutations: The number of arrangements of n objects taken r at a time is n! P(n, r) = n (n 1) (n r + 1) = (n r)!. Permutations of objects with some alike: The number of different permutations (arrangements), where order matters, of a set of n objects (taken n at a time) where r of the objects are identical is n! r!. Consider a set of n objects which is equal to the disjoint union of k subsets, A1, A2,..., Ak, of objects in which the objects in each subset Ai are identical and the objects in different subsets Ai and Aj, i j are not identical. Let ri denotes the number of objects in set Ai, then the number of different permutations of the n objects (taken n at a time) is n! r1!r2!... rn!. This can also be considered as an application of the technique of overcounting where we count a larger set and then divide. Combinations: The number of ways of choosing a subset of (or a sample of) r objects from a set with n objects, where order does not matter, is C(n, r) = P (n, r) r! n! = r!(n r)!. Note this was also an application of the technique of overcounting. 2 Problem Solving Strategy: You may be able to solve a counting problem with a single principle or a problem may be a multilevel problem requiring repeated application of one or several principles. When asked to count the number of objects in a set, it often helps to think of how you might complete the task of constructing an object in the set. It also helps to keep the technique of overcounting in mind. The following flowchart from your book may help you decide whether to use the multiplication principle, the addition rule, the formula for the number of permutations or the formula for the number of combinations for a problem or a problem part requiring one of these. 4 Example An experiment consists of rolling a 20 sided die three times. The number on top of each die is recorded. The numbers are written down in the order in which they are observed. How many possible ordered triples of numbers can result from the experiment? (Note the triple (17, 10, 3) is not the same result as the triple (3, 10, 17). ) There are 20 ways each throw can come up and the order is important so the answer is = 20 3 = 8000. 5 Example (Hoosier Lottery) When you buy a Powerball ticket, you select 5 different white numbers from among the numbers 1 through 59 (order of selection does not matter), and one red number from among the numbers 1 through 35. How many different Powerball tickets can you buy? If you check out the Powerball web site you will see that you need to select 5 distinct white numbers, so you can do this C(59, 5) = 5, 006, 386 ways. Then you can pick the red number C(35, 1) = 35 ways so the total number of tickets is C(59, 5) P(35, 1) = 5, 006, = 175, 223, 510. 6 Often problems fit the model of pulling marbles from a bag. For example many of our previous problems involving poker hands fit this model. Polling a population to conduct an observational study also fit this model. Example: An bag contains 15 marbles of which 10 are red and 5 are white. 4 marbles are selected from the bag. There s ambiguity here: e.g., if on one draw I select four red marbles, and on another draw I select a different four red marbles, are these considered the same sample or not? We ll assume that they are not the same sample. For example, we could imagine that the marbles are numbered, each with a different number, so that we can tell marbles of the same color apart. This way of thinking will be very useful for calculating probabilities later, when we try to set up an equally likely sample space 7 10 red, 5 white, numbered marbles (a) How many (different) samples (of size 4) are possible? The order does not matter but the numbers do so we are selecting 4 elements from a set of elements. Hence the answer is C(15, 4) = 1, 365. (b) How many samples (of size 4) consist entirely of red marbles? The order does not matter but the numbers do so we are selecting 4 elements from a set of 10 elements. Hence the answer is C(10, 4) = 210. 8 10 red, 5 white, numbered marbles (c) How many samples have 2 red and 2 white marbles? We can select 2 numbered red marbles in C(10, 2) ways and 2 numbered white marbles in C(5, 2) ways. Neither choice affects the other so the answer is C(10, 2) C(5, 2) = = 450. (d) How many samples (of size 4) have exactly 3 red marbles? We can select 3 numbered red marbles in C(10, 3) ways and 1 numbered white marble in C(5, 1) ways. Neither choice affects the other so the answer is C(10, 3) C(5, 1) = = 600. 9 10 red, 5 white, numbered marbles (e) How many samples (of size 4) have at least 3 red? The answer is the number of samples with 3 red plus the number of samples with 4 red. We can select 4 numbered red marbles in C(10, 4) ways and 0 numbered white marbles in C(5, 0) ways. Neither choice affects the other so the answer is C(10, 4) C(5, 0) = = 210. From the last example, there are 600 ways to select samples with exactly 3 red marbles so our answer is = 810. 10 10 red, 5 white, numbered marbles (f) How many samples (of size 4) contain at least one red marble? One answer is the number with exactly 1 + the number with exactly 2... the number with exactly 4. This is C(10, 1) C(5, 3)+C(10, 2) C(5, 2)+C(10, 3) C(5, 1)+C(10, 4) C(5, 0) which is = = 1, 360 It is also the total number of samples (1, 365) minus the number of samples with no red marbles which is C(10, 0) C(5, 4) = 5. 11 Example: Recall that a standard deck of cards has 52 cards. The cards can be classified according to suits or denominations. There are 4 suits, hearts, diamonds, spades and clubs. There are 13 cards in each suit. There are 13 denominations, Aces, Kings, Queens,...,Twos, with 4 cards in each denomination. A poker hand consists of a sample of size 5 drawn from the deck. Poker problems are often like urn problems, with a hitch or two. (a) How many poker hands consist of 2 Aces and 3 Kings? You can pick aces in C(4, 2) ways and kings in C(4, 3) ways. Neither choice affects the other so the answer is C(4, 2) C(4, 3) = 6 4 = 24. 12 There are 13 ways to pick the first denomination. Then are then C(4, 3)ways to pick 3 cards of that denomination. There are 12 ways to pick the second denomination and then C(4, 2) ways to pick 2 cards of that denomination. Hence there are 13 C(4, 3) 12 C(4, 2) = = 3, 744. Mixed Counting Problems (b) How many poker hands consist of 2 Aces, 2 Kings and a card of a different denomination? You can pick the 2 aces, 2 kings in C(4, 2) C(4, 2) = 6 6 = 36 ways. You can pick the remaining card in any of 52 8 = 44 ways so the answer is = 1, 584. (c) How many Poker hands have three cards from one denomination and two from another (a full house)? 13 (d) A royal flush is a hand consisting of an Ace, King, Queen, Jack and Ten, where all cards are from the same suit. How many royal flushes are possible? There is exactly 1 way to pick a royal flush in each suit so there are 4 of them. (e) A flush is a hand consisting of five cards from the same suit. How many different flushes are possible? There are C(13, 5) ways to get all cards of the same suit so there are C(13, 5) C(4, 1) = 1, = 5, 148 flushes. 14 Another useful model to keep in mind is that of repeatedly flipping a coin. This is especially useful for counting the number of outcomes of a given type when the experiment involves several repetitions of an experiment with two outcomes. We will explore probabilities for experiments of this type later when we study the Binomial distribution. We have already used this model in taxi cab geometry. Example: Coin Flipping Model If I flip a coin 20 times, I get a sequence of Heads (H) and tails (T). (a) How many different sequences of heads and tails are possible? There are 2 ways the first flip can come up; 2 more for the second and so on. Hence 2 2 = 2 20 = 1, 048, 576. 15 (b) How may different sequences of heads and tails have exactly five heads? Now we want to keep track of how many heads/tails there are in our sequence. This problem is similar to the taxi cab problem. There are 20 positions which need to be filled with either an 'H' or a 'T'. If we want exactly h heads in the sequence the answer if C(20, h). To see we are on the right track recall 2 n = C(n, 0) + C(n, 1) + C(n, 2) + C(n, 3) + + C(n, n) so the number of sequences with 0 heads plus the number of sequences with 1 head plus... plus the number of sequences with 20 heads is all the sequences so should be 2 20 as in part (a). The actual answer to our problem is C(20, 5) = 15, 504. 16 (c) How many different sequences have at most 2 heads? We did the work in part (b). The answer is C(20, 0) + C(20, 1) + C(20, 2) = = 211 (d) How many different sequences have at least three heads? C(20, 3) + C(20, 4) + + C(20, 19) + C(20, 20). OR 2 20 ( C(20, 0)+C(20, 1)+C(20, 2) ) = 1, 048, = 1, 048, 365 17 Example To make a non-vegetarian fajita at Lopez s Grill, you must choose between a flour or corn tortilla. You must then choose one type of meat from 4 types offered. You can then add any combination of extras (including no extras). The extras offered are fajita vegetables, beans, salsa, guacamole, sour cream, cheese and lettuce. How many different fajitas can you make? Think of this from the point of view of the kitchen. An order comes in and you need to assemble it. First you select the tortilla: 2 choices. Then you add the meat: 4 choices. So far there are 2 4 = 8 possibilities. Now you need to add the extras. There are 7 extras and the order can be any subset of them. Hence your choices are any subset of this set with 7 elements so 2 7 = 128. Hence the total possible is = 1024. 18 Extra Problems Example (a) How many different words (including nonsense words) can you make by rearranging the letters of the word EFFERVESCENCE E 5; F 2; R 1; V 1; S 1; C 2; N 1. Hence there are = 13 letters total and so there are 13! 2! 5! 2! 1! 1! 1! 1! = P(13, 5) 4 = 51, 891, = 12, 972, 960 words. 19 Extra Problems (b) How many different 4 letter words (including nonsense words) can be made from the letters of EFFERVESCENCE, if letters cannot be repeated? There are 7 distinct letters so if repetitions are not permitted the answer is P(7, 4) = 840. (c) How many different 4 letter words (including nonsense words) can be made from the letters of the above word, if letters can be repeated? Answer: 7 4. Do not confuse this with the MUCH harder problem of given 13 tiles with the letters in EFFERVESCENCE, how many 4 letter words can be produced? So for example, you could use F twice but not 3 times. 20 Extra Problems Example The Notre Dame Model UN club has 20 members. Five are seniors, four are juniors, two are sophomores and nine are freshmen. (a) In how many ways can the club select a president, a secretary and a treasurer if every member is eligible for each position and no member can hold two positions? 20 members, 3 officers so P(20, 3). Note you are selecting an ordered subset of 3 distinct elements. (b) In how many ways can the club choose a group of 5 members to attend the next Model UN meeting in Washington. Answer: C(20, 5). This time you need a subset of all the members which has 5 elements but the order isn t important. 21 Extra Problems (c) In how many ways can the club choose a group of 5 members to attend the next Model UN meeting in Washington if all members of the group must be freshmen? Answer: C(9, 5) since you now must select your subset from the set of 9 freshmen. 22 Extra Problems (d) In how many ways can the group of five be chosen if there must be at least one member from each class? There are 5 ways to select a senior, 4 ways to select a junior, 2 ways to select a sophomore and 9 ways to select a freshman. This gives = 360 ways to select a subset with 4 elements containing one member of each class. When you have done this there are 20 4 = 16 members left and you may choose any one of these to round out the group. Hence the answer is = 2, 880. You 2 must divide by 2 because each set of 5 elements selected by this procedure occurs twice. 23 Extra Problems Here is another approach. Because there are 5 members in the subset and 4 classes, exactly one class occurs twice. If there are 2 seniors, these can be selected in C(5, 2) ways and the set filled out with 1 junior, 1 sophomore and 1 freshman, hence in C(5, 2) = 720 ways. If there are 2 juniors, these can be selected in C(4, 2) ways and the set filled out with 1 senior, 1 sophomore and 1 freshman, hence in 5 C(4, 2) 2 9 = 540 ways. If 2 sophomores, 5 4 C(2, 2) 9 = 180. If 2 freshmen, C(9, 2) = 1, 440. Hence the answer is , 440 = 2, 880. 24 Extra Problems Example Harry Potter s closet contains 12 numbered brooms, of which 8 are Comet Two Sixty s (numbered 1-8) and 4 are Nimbus Two Thousand s (Numbered 9-12). Harry, Ron, George and Fred want to sneak out for a game of Quidditch in the middle of the night. They don t want to turn on the light in case Snape catches them. They reach in the closet and pull out a sample of 4 brooms. 25 Extra Problems (a) How many different samples are possible? This is not a well-defined question. Do you want to know how many different sets of brooms you can get or do you want to know how many ways there are if we keep track of which broom Harry gets, which one Ron gets, and so on. In other words, do you want subsets or ordered subsets? For subsets, the answers is C(12, 4) = 495; for ordered subsets the answer is P(12, 4) = 11, 880. 26 Extra Problems (b) How many samples have only Comet Two Sixty s in them? Replace the 12 in the answers for part (a) with 8. (c) How many samples have exactly one Comet Two Sixty in them? The unordered version solution is familiar. There are C(8, 1) = 8 ways to pick the Comet Two Sixty and C(4, 3) = 4 ways to pick the rest so the answer is 8 4 = 32. To do the ordered version, observe that once you have an unordered set of 4 elements, there are 4! = 24 ways to order it. Hence the ordered answer is = 768. 27 Extra Problems (d) How many samples have at least 3 Comet Two Sixty s? Figure out how many samples there are with exactly 3; then figure out how many there are with exactly 4 and then add the two answers. For exactly k Comet Two Sixty s we have C(8, k) C(4, 4 k) unordered subsets and therefore C(8, k) C(4, 4 k) 4! ordered ones.

Constants Sentence Examples Here there are two arbitrary constants, which may be adjusted in various ways. It is well known that singly, doubly and trebly linked carbon atoms affect the physical properties of substances, such as the refractive index, specific volume, and the heat of combustion; and by determining these constants for many substances, fairly definite values can be assigned to these groupings. If we express the pressure, volume and temperature as fractions of the critical constants, then, calling these fractions the " reduced " pressure, volume and temperature, and denoting them by 7r, 0 and 0 respectively, the characteristic equation becomes (7+3/0 2)(30-i) =80; which has the same form for all substances. Obviously, therefore, liquids are comparable when the pressures, volumes and temperatures are equal fractions of the critical constants. An important connexion between heats of combustion and constitution is found in the investigation of the effect of single, double and triple carbon linkages on the thermochemical constants.Advertisement The difference of potential between two solutions of a substance at different concentrations can be calculated from the equations used to give the diffusion constants. Hermite expresses the quintic in a forme-type in which the constants are invariants and the variables linear covariants. Hesse showed independently that the general ternary cubic can be reduced, by linear transformation, to the form x3+y3+z3+ 6mxyz, a form which involves 9 independent constants, as should be the case; it must, however, be remarked that the counting of constants is not a sure guide to the existence of a conjectured canonical form. Thus the ternary quartic is not, in general, expressible as a sum of five 4th powers as the counting of constants might have led one to expect, a theorem due to Sylvester. When the magnetizing current is twice reversed, so as to complete a cycle, the sum of the two deflections, multiplied by a factor depending upon the sectional area of the specimen and upon the constants of the apparatus, gives the hysteresis for a complete cycle in ergs per cubic centimetre.Advertisement He applied his method with good effect, however, in testing a large number of commercial specimens of iron and steel, the magnetic constants of which are given in a table accompanying his paper. The field due to a coil can be made as nearly uniform as we please throughout a considerable space; its intensity, when the constants of the coil are known, can be calculated with ease and certainty and may be varied at will'through wide ranges, while the apparatus required is of the simplest character and can be readily constructed to suit special purposes. In nickel the maximum change of the elastic constants is remarkably large, .amounting to about 15% for Young's modulus and 7% for rigidity; with increasing fields the elastic constants first decrease and then increase. In a 29% nickel-steel, magnetization increases the constants by a small amount. Rowland,' whose careful experiments led to general recognition of the fact previously ignored by nearly all investigators, that magnetic susceptibility and permeability are by no means constants (at least in the case of the ferromagnetic metals) but functions of the magnetizing force.Advertisement The final achievement of Lagrange in this direction was the extension of the method of the variation of arbitrary constants, successfully used by him in the investigation of periodical as well as of secular inequalities, to any system whatever of mutually interacting bodies.' He also devoted much attention to the pyroelectric phenomena of crystals, which served as the theme of one of the two memoirs he presented for the degree of D.Sc. in 1869, and to the determination of crystallographic constants. The raw materials are selected with great care to assure chemical purity, but whereas in most glasses the only impurities to be dreaded are those that are either infusible or produce a colouring effect upon the glass, for optical purposes the admixture of other glass-forming bodies than those which are intended to be present must be avoided on account of their effect in modifying the optical constants of the glass. The simplest method of determining it numerically is, therefore, that adopted by Faraday.4 Table Dielectric Constants (K) of Solids (K for Air = I). Here is carried out the work of standardizing measuring instruments of various sorts in use by manufacturers, the determination of physical constants and the testing of materials.Advertisement If suitable values are chosen for these constants, the formula can be made to represent the dispersion of ordinary transparent media within the visible spectrum very well, but when extended to the infra-red region it often departs considerably from the truth, and it fails altogether in cases of anomalous dispersion. The equations finally arrived at are DX2(A2_ 2) (x2_ A2m)2+g2A2 ' DgA3 (A A l m) 2 +g 2 A2 ' where is the wave-length in free ether of light whose refractive index is n, and A m the wave-length of light of the same period as the electron, is a coefficient of absorption, and D and g are constants. The form of the limacon depends on the ratio of the two constants; if a be greater than b, the curve lies entirely outside the circle; if a equals b, it is known as a cardioid; if a is less than b, the curve has a node within the circle; the particular case when b= 2a is known as the trisectrix. The constant a has the same value I 2 for crown and flint glass, so that there are only three disposable constants left. If within the range5100-3700A, the constants are determined once for all, the formula seems capable of giving by interpolation results accurate to o 2 A, but as a rule the range to which the formula is applied will be much less with a corresponding gain in the accuracy of the results.Advertisement A homogeneous oscillation is one which for all time is described by a circular function such as sin(nt+ a), t being the time and n and a constants. The first of the forms which contains three disposable constants did good service in the hands of their authors, but breaks down in important cases when odd powers of s have to be introduced in addition to the even powers. The second form contains two or three constants according as N is taken to have the same value for all elements or not. It then possesses four adjustable constants, and more can therefore be expected from it. As he takes N to be strictly the same for all elements the equation has only three disposable constants A, a and b. This form has the advantage that the constants of the equation when applied to the spectra of the alkali metals show marked regularities. If we compare Balmer's formula with the general equation of Ritz, we find that the two can be made to agree if the ordinary hydrogen spectrum is that of the side branch series and the constants a', b, c and d are all put equal to zero.' If s represents the series of integer numbers the distribution of frequency may be represented by C+Bs2, where C and B are constants. If we wish to be more general, while still adhering to Deslandres' law as a correct representation of the frequencies when s is small, we may write n - A (s+ 1 1) 2 - - a Po+Pi(s + c) -F +pr(s+ c)r' where s as before represents the integer numbers and the other quantities involved are constants. A band might in that case fade away towards zero frequencies, and as s increases, return again from infinity with diminishing distances, the head and the tail pointing in the same direction; or with a different value of constants a band might fade away towards infinite frequencies, then return through the whole range of the spectrum to zero frequencies, and once more return with its tail near its head. But, on the other hand, no one pretends to have found the rigorous expression for the law, and the appropriate approximation may take quite different forms when constants which are large in one case are small in the other. Omitting correction terms depending on the temperature and on the inductive effect of the earth's magnetism on the moment of the deflecting magnet, if 0 is the angle which the axis of the deflected magnet makes with the meridian when the centre of the deflecting magnet is at a distance r, then zM sin B=I+P+y2 &c., in which P and Q are constants depending on the dimensions and magnetic states of the two magnets. The value of the constants P and Q can be obtained by making deflexion experiments at three distances. Attempts have been made to co-ordinate this ionizing power of solvents with their dielectric constants, or with their chemical properties. The physical constants associated with the name will scarcely be changed, since the proportion of the "companions" is so small. Main as chief assistant at the Royal Observatory, Greenwich, and at once undertook the fundamental task of improving astronomical constants. Every time, therefore, that a speculum is repolished, the future quality of the instrument is at stake; its focal length will probably be altered, and thus the value of the constants of the micrometer also have to be redetermined. It is not possible to deduce a more satisfactory value from the latent heat and the change of density, because these constants are very difficult to determine. That of Biot is far more complicated and troublesome, but admits greater accuracy of adaptation, as it contains five constants (or six, if 0 is measured from an arbitrary zero). The omission of the additive arbitrary constants of integration in (8) is equivalent to a special choice of the origin 0 of co-ordinates; viz. To obtain the complete solution of (II) we must of course superpose the free vibration (6) with its arbitrary constants in order to obtain a complete representation of the most general motion consequent on arbitrary initial conditions. The resulting Z+R equations are not as a rule easy of application, owing to the fact that the moments and products of inertia A, B, C, F, G, H are not constants but vary in conse- 0 quence of the changing orientation of the body with respect to the co-ordinate axes. On assuming the directorship of the Nautical Almanac he became very strongly impressed with the diversity existing in the values of the elements and constants of astronomy adopted by different astronomers, and the injurious effect which it exercised on the precision and symmetry of much astronomical work. A valuable summary of a considerable part of this work, containing an account of the methods adopted, the materials employed, and the resulting values of the various quantities involved, was published in 1895, as a supplement to the American Ephemeris for 1897, entitled The Elements of the Four Inner Planets and the Fundamental Constants of Astronomy. At the international conference, which met at Paris in 1896 for the purpose of elaborating a common system of constants and fundamental stars to be employed in the various national ephemerides, Newcomb took a leading part, and at its suggestion undertook the task of determining a definite value of the constant of precession, and of 1 Lionville, t. At the time when Maxwell developed his theory the dielectric constants of only a few transparent insulators were known and these were for the most part measured with steady or unidirectional electromotive force. Experimental methods were devised for the further exact measurements of the electromagnetic velocity and numerous determinations of the dielectric constants of various solids, liquids and gases, and comparisons of these with the corresponding optical refractive indices were conducted. He investigated the optical constants of the eye, measured by his invention, the ophthalmometer, the radii of curvature of the crystalline lens for near and far vision, explained the mechanism of accommodation by which the eye can focus within certain limits, discussed the phenomena of colour vision, and gave a luminous account of the movements of the eyeballs so as to secure single vision with two eyes. If, in the first place, monochromatic aberrations be neglected - in other words, the Gaussian theory be accepted - then every reproduction is determined by the positions of the focal planes, and the magnitude of the focal lengths, or if the focal lengths, as ordinarily happens, be equal, by three constants of reproduction. If all three constants of reproduction be achromatized, then the Gaussian image for all distances of objects is the same for the two colours, and the system is said to be in " stable achromatism." If r be the number of quotients in the recurring cycle, we can by writing down the relations connectin g the successive p's and q's obtain a linear relation connecting p nr +m, t'(n-1)r +m, +m in which the coefficients are all constants. The work falls into two parts, which treat of the asymptotes and singularities of algebraical curves respectively; and extensive use is made of the method of counting constants which plays so large a part in modern geometrical researches. The points in question have since been called (it is believed first by Dr George Salmon) the circular points at infinity, or they may be called the circular points; these are also frequently spoken of as the points I, J; and we have thus the circle characterized as a conic which passes through the two circular points at infinity; the number of conditions thus imposed upon the conic is = 2, and there remain three arbitrary constants, which is the right number for the circle. The expression 2 is that of the number of the disposable constants in a curve of the order m with nodes and cusps (in fact that there shall be a node is I condition, a cusp 2 conditions) and the equation (9) thus expresses that the curve and its reciprocal contain each of them the same number of disposable constants. The determination of the constants in Gauss's theory of terrestrial magnetism occupied him at intervals for over forty years. In the language of algebra putting m l, m2, m 3, &c. for the masses of the bodies, r1.2 r1.3 r2.3, &c. for their mutual distances apart; vi, v 2, v 3, &c., for the velocities with which they are moving at any moment; these quantities will continually satisfy the equation orbit, appear as arbitrary constants, introduced by the process of integration. In a case like the present one, where there are two differential equations of the second order, there will be four such constants. The result of the integration is that the co-ordinates x and y and their derivatives as to the time, which express the position, direction of motion and speed of the planet at any moment, are found as functions of the four constants and of the time. Putting a, b, c, d, for the constants, the general form of the solution will be x = fl (a,b,c,d,t) y = f2(a,b,c,d,t) From these may be derived by differentiation as to t the velocities dt =f '1(a,b,c,d,t) = x' ? The arbitrary constants, a, b, c and d, are the elements of the orbit, or any quantities from which these elements can be obtained. This fact is fully expressed by the equations (4) where we have constants on one side of the equation equal to functions of the variables on the other. Not less comprehensive has been the work carried out by Professor Newcomb of raising to a higher grade of perfection, and reducing to a uniform standard, all the theories and constants of the solar system. The determination of the curves of constant retardation depends upon expressing the retardation in terms of the optical constants of the crystal, the angle of incidence and the azimuth of the plane of incidence. The optical constants (refractive index and co-efficient of extinction) of the metal may then be obtained from observations of the principal incidence and the elliptic polarization then produced. One method consists in finding directly the elliptic constants of the vibration by means of a quarterwave plate and an analyser; but the more usual plan is to measure the relative retardation of two rectangular components of the stream by a Babinet's compensator. This is the equation to a parabola, and is equivalent to the empirical formula of Avenarius, with this difference, that in Tait's formula the constants have all a simple and direct interpretation in relation to the theory. It simplifies the theory, and gives a possible relation between the constants, but it does not appear to remove the complication above referred to, which seems to be inseparable from any conduction theory. They were constants in a world where humans and their inventions passed through the world, less significant than an exhaled breath. The class that defines the constants that are used to identify generic SQL types, called JDBC types. I am using the modification of the Eshelby approach to model for the elastic constants of porous coatings. Just as physical constants provide " boundary conditions " for the physical universe, mathematical constants somehow characterize the structure of mathematics. These are TAB format, byte-length specifiers and Hollerith constants. Curiously, however, the basic properties of these materials (band gap, elastic constants, piezoelectric constants) are not well known. These differences arise because one or more stability constants has a bad value or indeed should not be present or should not be included. Quarks Most uses of the resource manager involve defining names, classes, and representation types as string constants. In biological systems the reference values are often encoded genetically, in the binding constants of proteins for their allosteric effectors. It is not, generally, possible to use hexadecimal or decimal constants in this fashion. Consider the language obtained from 1 K = by adding a denumerably infinite stock of new individual constants c 0, c 1, . The wavelength associated with the vector resultant of these three orthogonal propagation constants is just the free space wavelength lambda. A public constructor for this class has been purposely omitted and applications should use one of the constants from this class. The time constants and relative weightings yielded temporal window functions that heavily emphasize information occurring within the very temporal center of the window. I am analyzing synchrotron X-ray diffraction patterns and to work out the diffractometer constants I refined a silicon pattern. For the few cases where data are available - data, however, belonging to engines representing standard practice in their construction and in the design of cylinders and steam ports and passages - the law connecting p and v is approximately linear and of the form p=c - bv (28) where b and c are constants. Kauffmann (Ber., 1906, 39, p. 1 959) attempted an evaluation of the effects of auxochromic groups by means of the magnetic optical constants. In that case the main branch is found to represent the new series if a' and b 1 are also put equal to zero, so that n l r I I N = 4 - y2' where r takes successively the values 1.5, 2.5, 3.5 A knowledge of the constants now determines the trunk series, which should be n I I N - (I,5)2 The least refrangible of the lines of this series should have a wavelength 4687.88, and a strong line of this wave-length has indeed been found in the spectra of stars which are made up of bright lines, as also in the spectra of some nebulae. Poisson's application to them in 1809 of Lagrange's theory of the variation of constants; Philippe de Pontecoulant successfully used in 1829, for the prediction of the impending return of Halley's comet, a system of " mechanical quadratures " published by Lagrange in the Berlin Memoirs for 1778; and in his Theorie analytique du systeme du monde (1846) he modified and refined general theories of the lunar and planetary revolutions. The analysis is extended to a consideration of rate constants for chemical reactions between solutes in solution. The range column shows the range of specifiable values in the system constants. Static variables can be used to emulate constants, values that do n't change. I am analyzing synchrotron x-ray diffraction patterns and to work out the diffractometer constants I refined a silicon pattern. Only the constants of the excited triplet state may be varied. Your underhanded attempt to be glib and smarmy does n't undermine or change those historical constants. While Bob and Jillian have been the constants on the show, there have been other trainers involved as well. There are notable differences, but some constants. Silbermann, whose chief theoretical achievement was the recognition that the heat of neutralization of acids and bases was additively composed of two constants, one determined by the acid and the other by the base. But, at the same time, the constants in the above relation are not identical with those in the corresponding relation empirically deduced from observations on fatty hydrocarbons; and we are therefore led to conclude that a benzene union is considerably more stable than an ethylene union. Eliminating a and b between these relations, we derive P k V k /Tk= 8R, a relation which should hold between the critical constants of any substance. By actual observations it has been shown that ether, alcohol, many esters of the normal alcohols and fatty acids, benzene, and its halogen substitution products, have critical constants agreeing with this originally empirical law, due to Sydney Young and Thomas; acetic acid behaves abnormally, pointing to associated molecules at the critical point. From the relation between the critical constants Pk Vk/Tk = 37 R or T k /P k = 3 . It contains four independent constants; two of these may be calculated from the heats of combustion of saturated hydrocarbons, and the other two from the combustion of hydrocarbons containing double and triple linkages. In the article Crystallography the nature and behaviour of twinned crystals receives full treatment; here it is sufficient to say that when the planes and axes of twinning are planes and axes of symmetry, a twin would exhibit higher symmetry (but remain in the same crystal system) than the primary crystal; and, also, if a crystal approximates in its axial constants to 'a higher system, mimetic twinning would increase the approximation, and the crystal would be pseudo-symmetric. In such crystals each component plays its own part in determining the physical properties; in other words, any physical constant of a mixed crystal can be calculated as additively composed of the constants of the two components. Let s be the area of a single turn of the standard coil, n the number of its turns, and r the resistance of the circuit of which the coil forms part; and let S, N and R be the corresponding constants for a coil which is to be used in an experiment. In nickel-steels containing about 50 and 70% of nickel the maximum increase of the constants is as much as 7 or 8%. The hypothesis that the state was steady, so that interchanges arising from convection and collisions of the molecules produced no aggregate result, enabled him to interpret the new constants involved in this law of distribution, in terms of the temperature and its spacial differential coefficients, and thence to express the components of the kinetic stress at each point in the medium in terms of these quantities. Further Physical Properties of Sea-water.---The laws of physical chemistry relating to complex dilute solutions apply to seawater, and hence there is a definite relation between the osmotic pressure, freezing-point, vapour tension and boiling-point by which when one of these constants is given the others can be calculated. Van der Waal's equation (p-I- a/v 2) (v - b) = RT contains two constants a and b determined by each particular substance.

- Reactance (electronics) Reactance is a circuit element's opposition to an alternating current, caused by the build up of electric or magnetic fields in the element due to the current. Both fields act to produce counter emf that is proportional to either the rate of change (time derivative), or accumulation (time integral) of the current. In vector analysis, Reactance is the imaginary partof electrical impedance, used to compute amplitude and phase changes of sinusoidal alternating currentgoing through the circuit element. It is denoted by the symbol . The SI unit of reactance is the ohm. Both reactance and resistance are required to calculate the impedance , although in some circuits one of these may dominate: an approximate knowledge of the minor component is useful to determine if it may be neglected. The magnitude is the ratio of the voltageand current amplitudes, while the phase is the voltage–current phase difference. The reciprocal of reactance is Determining the voltage-current relationship requires knowledge of both the resistance and the reactance. The reactance on its own gives only limited physical information about an electrical component or network. # A positive reactance implies that the circuit is inductive, where phase of the voltage "leads" the phase of the current; while a negative reactance implies that the circuit is capacitive, where phase of the voltage "lags" the phase of the current # A reactance of zero implies the current and voltage are in phase and conversely if the reactance is non-zero then there is a phase difference between the voltage and current There are certain specific effects that depend on the reactance alone, for example; resonance in an series RLC circuitoccurs when the reactances "XC" and "XL" are equal but opposite, and the impedance has a phase angle of zero. Capacitive reactance is inversely proportionalto the signal frequencyand the capacitance . A capacitor consists of two conductors separated by an insulator, also known as a At low frequencies a capacitor is open circuit, as no current flows in the dielectric. A DC voltage applied across a capacitor causes charge to accumulate on one side; the electric fielddue to the accumulated charge is the source of the opposition to the current. When the potentialassociated with the charge exactly balances the applied voltage, the current goes to zero. Driven by an AC supply, a capacitor will only accumulate a limited amount of charge before the potential difference changes sign and the charge dissipates. The higher the frequency, the less charge will accumulate and the smaller the opposition to the current. Inductive reactance is proportionalto the signal frequencyand the inductance . An inductor consists of a coiled conductor. Faraday's law of electromagnetic induction gives the back emf (voltage opposing current) due to a rate-of-change of magnetic flux densitythrough a current loop. For an inductor consisting of a coil with loops this gives. The back-emf is the source of the opposition to current flow. A constant direct currenthas a zero rate-of-change, and sees an inductor as a short-circuit(it is typically made from a material with a low resistivity). An alternating currenthas a time-averaged rate-of-change that is proportional to frequency, this causes the increase in inductive reactance with frequency. The phase of the voltage across a purely reactive device (a device with a resistance of zero) "lags" the current by radians for a capacitive reactance and "leads" the current by radians for an inductive reactance. Note that without knowledge of both the resistance and reactance we cannot determine the voltage--current relationships. The origin of the different signs for capacitive and inductive reactance is the phase factor in the impedance. For a reactive component the sinusoidal voltage across the component is in quadrature (a phase difference) with the sinusoidal current through the component. The component alternately absorbs energy from the circuit and then returns energy to the circuit, thus a pure reactance does not dissipate power. # Pohl R. W. "Elektrizitätslehre." – Berlin-Gottingen-Heidelberg: Springer-Verlag, 1960. # Popov V. P. "The Principles of Theory of Circuits." – M.: Higher School, 1985, 496 p. (In Russian). # Küpfmüller K. "Einführung in die theoretische Elektrotechnik," Springer-Verlag, 1959. * [http://www.magnet.fsu.edu/education/tutorials/java/inductivereactance/index.html Interactive Java Tutorial on Inductive Reactance] National High Magnetic Field Laboratory * [http://www.geocities.com/SiliconValley/2072/elecrri.htm Resistance, Reactance, and Impedance] * [http://www.sweethaven.com/sweethaven/ModElec/dcac/SunShine/LessonMain01.asp?uNum=221 Inductive Reactance: Endless Examples & Exercises ] Wikimedia Foundation. 2010. Look at other dictionaries: Reactance (disambiguation) — Reactance may refer to:* Reactivity (chemistry), the rate at which a chemical substance tends to undergo a chemical reaction * Reactance (electronics), a circuit s opposition to alternating current * Reactance (psychology), an emotional reaction… … Wikipedia reactance — /ree ak teuhns/, n. 1. Elect. the opposition of inductance and capacitance to alternating current, expressed in ohms: equal to the product of the sine of the angular phase difference between current and voltage and the ratio of the effective… … Universalium List of electronics topics — Alphabetization has been neglected in some parts of this article (the b section in particular). You can help by editing it. This is a list of communications, computers, electronic circuits, fiberoptics, microelectronics, medical electronics,… … Wikipedia Radio electronics — * For the magazine, see Radio Electronics Radio electronics is the sub field of electrical engineering concerning itself with the class of electronic circuits which receive or transmit radio signals.Typically such circuits must operate at radio… … Wikipedia Stub (electronics) — In electronic signaling, a stub is a section of transmission line with a calculated length. It is used for impedance matching or to obtain an exact capacitance or inductance.Because stubs take on reactive properties as a function of their… … Wikipedia Choke (electronics) — For other uses, see Choke (disambiguation). A choke, with two 47mH windings and rated to handle 0.6A A choke is a coil of insulated wire, often wound on a magnetic core, used as a passive inductor which blocks higher frequency alternating current … Wikipedia Capacitor — This article is about the electronic component. For the physical phenomenon, see capacitance. For an overview of various kinds of capacitors, see types of capacitor. Capacitor Modern capacitors, by a cm ruler Type Passive … Wikipedia Inductor — An inductor is a passive electrical component designed to provide inductance in a circuit.Inductors store energy in a magnetic field created when an electric current flows through them. They are usually implemented by some sort of coiled… … Wikipedia Reactive — can refer to:*Generally, capable of having a reaction *Reactance (electronics), the imaginary component of AC impedance *In behavioral medicine reactive often refers to a treatment approach where a does not initiate contact as opposed to… … Wikipedia parametric amplifier — Electronics. a device, as an electron tube or transistor, that amplifies a high frequency input signal by sinusoidally varying the reactance of the circuit. [1955 60] * * * … Universalium